- 
 
 IA11 
 
 Southern Branch 
 of the 
 
 University of California 
 
 Los Angeles 
 
 Form L I
 
 This book is DUE on the last date stamped below 
 
 ' ---- 
 21 193o 
 
 2 
 
 8 
 
 MAR 26 1931 
 APR 28 1931 
 
 MAY i 2 193* 
 ' APR 29 19ST
 
 THE PRESENT TEACHING OF 
 MATHEMATICS IN GERMANY 
 
 By 
 DAVID EUGENE SMITH 
 
 WITH THE CO-OPERATION OF VARIOUS 
 GRADUATE STUDENTS 
 
 STATE NOEMAL SCHOOL 
 
 LOS ANGELES. 
 
 PUBLISHED BY 
 
 Srarhrrs GJuUrrjr, (Enlttmbia 'iluturnntg 
 
 NEW YORK CITY 
 
 1912
 
 CONTENTS 
 
 Chapter Page 
 
 * I. GERMAN vs. AMERICAN CONDITIONS . David Eugene Smith i 
 
 ^ II. EVOLUTION OF THE REFORM IN GERMANY . Isidore Skolnick 12 
 
 III. THE SECONDARY SCHOOLS OF HESSE AND BADEN . . .18 
 
 Miriam E. West 
 
 IV. THE SECONDARY SCHOOLS OF THE HANSEATIC STATES . .25 
 
 Katharine S. Arnold and Ruth Fitch Cole 
 
 V. THE SECONDARY SCHOOLS OF WURTEMBERG . Isidore Skolnick 32 
 
 VI. THE SECONDARY SCHOOLS OF BAVARIA . M. J. Leventhal 39 
 
 VII. THE HIGHER SCHOOLS FOR BOYS IN PRUSSIA . .44 
 
 Robert King Atwell 
 VIII. THE SECONDARY SCHOOLS OF ELSASS AND LOTHRINGEN . . 50 
 
 Maurice Levine 
 '" IX. MATHEMATICS IN GERMAN TECHNICAL SCHOOLS ... 63 
 
 Donald T. Page 
 
 X. THE GERMAN MIDDLE TECHNICAL SCHOOLS . Miriam E. West 66 
 'XI. MATHEMATICS IN THE GERMAN SCHOOLS OF NAVIGATION . 75 
 
 Donald T. Page 
 
 XII. COMMERCIAL PROBLEMS IN THE HIGHER SCHOOLS OF GERMANY 78 
 
 W. F. Enteman 
 
 XIII. MATHEMATICS IN THE TEXT-BOOKS ON PHYSICS . A. T. French 90 
 
 XIV. GOVERNMENT EXAMINATIONS IN PRUSSIA AND THE NORTH GER- 
 
 MAN STATES . Cilda Langfitt Smith and Katherine Simpson 97 
 
 XV. DESCRIPTIVE GEOMETRY IN THE REALSCHULEN . . . 106 
 
 Louise Eugenie Harvey and Jessie Mae Reynolds 
 
 XVI. CONCLUSION . . Eleanora T. Miller 121
 
 <SA \\ 
 
 TEACHERS COLLEGE RECORD 
 
 Vol. XIII MARCH, 1912 No. 2 
 
 PREFACE 
 
 In one of the graduate classes of Teachers College it is the 
 custom to devote part of each year to a study of educational 
 problems in the field of mathematics in other countries. As a 
 rule the students in this class are teachers of some experience. 
 All of them are college graduates and all are desirous of know- 
 ing the best that the world is doing in the teaching of their 
 chosen subject. In this desire they are encouraged in every prac- 
 tical way, it being the position of the department that such 
 knowledge is one of the two foundation stones upon which they 
 must build, the other being a sound knowledge of the subject 
 matter. With these two must, of course, go much else, a 
 knowledge of how education has come to be what it is, an out- 
 look into modern tendencies, a knowledge of the laws of mind, 
 and so on ; but without a knowledge of mathematics and a knowl- 
 edge of how it has been and is being taught at its best, all the 
 rest lacks application and develops the mere experimenter (and 
 too often an. opinionated one) in this important field. 
 
 It happens that during the current year the German branch 
 of the International Commission on the Teaching of Mathe- 
 matics has issued more reports than that of any other country. 
 All of the most important nations, including our own, are at 
 present issuing these reports, but Germany has thus far excelled 
 all others both in the range of the investigation and in the 
 promptness with which the results have been published. On 
 
 -67] I
 
 2 Teachers College Record [68 
 
 this account the present class has given more attention to mathe- 
 matics in Germany than to that in other countries, although 
 covering a considerable range of work in France, Holland, Eng- 
 land, Sweden, Russia, Italy, America, and Austria. The results 
 of some of this investigation have been summarized by the vari- 
 ous members of the class and constitute the body of this work. 
 
 It has been impossible to allow enough space to the various 
 reports to permit of many details as to the schools, the courses, 
 the pupils, the teaching staff, and the methods employed. Such 
 details could hardly be looked for in a condensed statement of 
 this kind. All that has been attempted in these reports is a bird's- 
 eye view of the field, and this is all that could be expected. The 
 details have been presented and discussed in the class, but the 
 reader who wishes for more specific information will have to 
 consult the publications in their original form. These are all 
 available at a reasonable price, and may, like those of other 
 countries, be secured by addressing Messrs. Georg et Cie, Edi- 
 teurs, Geneva, Switzerland. If, through their circulation and 
 through such abstracts as those which make up this publication, 
 we can awaken to the large questions in the teaching of mathe- 
 matics, the work of the International Commission will prove most 
 salutary. To get away from such narrow ideas as that mathe- 
 matical problems must all be physical problems, and that geometry 
 fails except as it relates to the workshop; to divorce ourselves 
 from the misconceptions that a country like Germany or France 
 no longer teaches algebra and geometry as separate and distinct 
 subjects; to come to the state of forming opinions only after 
 finding what others are doing, these are some of the fortunate 
 results that may be anticipated from such study. 
 
 With respect to Germany, it should first be understood that 
 the schools of that country are doing certain kinds of work that 
 we are not doing, some that we cannot do under present condi- 
 tions, and some that we might not care to do if we could. On 
 the other hand they are doing certain work that we wish we 
 might do, and that, in due time, we shall probably come to do 
 as well as it is being done there. Some of the reasons for the 
 difference between the work in Germany and that in our country 
 are set forth in Chapter I and several desirable lines of improve-
 
 69] The Present Teaching of Mathematics in Germany 3 
 
 ment in our own work will be suggested in the chapters that 
 follow. 
 
 It will be observed that these reports enter but little into the 
 description of the schools themselves. The reader who wishes 
 to obtain information of this kind, including a statement of 
 the methods of conducting the classes in the various types of 
 schools, may consult the excellent little volume prepared a few 
 years ago by Professor J. W. A. Young, " Mathematics in the 
 vSchools of Prussia" (New York, Longmans). 
 
 For the benefit of the reader who is unfamiliar with the Ger- 
 man school system, a brief statement of the character and scope 
 of the work of the secondary schools is necessary before begin- 
 ning the various chapters. The three leading types of secondary 
 schools are as follows : the Gymnasium, with both Latin and 
 Greek ; the Realgymnasium, with Latin but no Greek ; and the 
 Oberrealschule, with neither Latin nor Greek. The course ex- 
 tends over a period of nine years and the pupil must, in general, 
 have attained the age of nine years before he can enter. The 
 classes are named as follows, beginning with the lowest; Sexta 
 (VI), Quinta (V), Quarta (IV), Untertertia (UIII), Obertertia 
 (OIII), Untersecunda (UII), Obersecunda (Oil), Unterprima 
 (UI), and Oberprima (OI). 
 
 There is another group of schools offering a six-year course, 
 the work corresponding exactly to the first six years of the nine- 
 year schools. These are the Progymnasium, the Realprogym- 
 nasium, and the Realschule. 
 
 None of these institutions corresponds exactly to any Amer- 
 ican school and each is consequently referred to in the subse- 
 quent chapters by the German name. 
 
 The labor of editing the several monographs that make up this 
 volume has fallen chiefly upon Miss Eleanora T. Miller, who 
 also wrote the concluding chapter, and to her, as well as to the 
 various contributors, are due the thanks of the department. 
 
 DAVID EUGENE SMITH.
 
 CHAPTER I 
 
 2 7 S /o 
 GERMAN VERSUS AMERICAN CONDITIONS 
 
 David Eugene Smith 
 
 Before taking up the special reports on mathematics in the 
 schools of Germany, it is well to consider briefly one question 
 that constantly occurs to American teachers who seriously seek, 
 after studying the subject, to improve the mathematics taught in 
 our country. The question is a very natural one, viz. : Why can 
 we not, in the same number of years, cover as wide a field of 
 mathematics as the Germans? Their Gymnasium, for example, 
 has a nine-year course, following a minimum of three years in 
 the elementary school. Here are twelve years, often lengthened 
 to thirteen, however, to be placed alongside the time needed to 
 complete the work through our high school. And yet, in some 
 parts of Europe, and in particular of Germany, students in such 
 schools study not merely algebra and geometry as with us, but 
 also trigonometry, descriptive geometry, geometric drawing, a 
 little modern geometry of a simple kind, some analytics, some 
 calculus, and a fair amount of mathematical mechanics. What 
 we wish to know is, are they really doing this ? and, if so, what 
 is their curriculum? how is the work arranged? and why are we 
 not doing as well ? 
 
 The question as to what they are doing in Germany is briefly 
 answered in the subsequent chapters. The types of schools, the 
 courses of study, the general range of work, the preparation 
 of teachers, something of the methods employed, and the nature 
 of the problems, are all set forth with sufficient detail to give 
 a fair reply to the question. But it was not the problem set 
 for the writers to consider the reasons for the difference between
 
 6 Teachers College Record [72 
 
 Germany and America in this respect, and hence a brief dis- 
 cussion of this question is in order at this time. 
 
 In the first place, the climate of Germany is thought to allow 
 for a longer school year than with us. How much of this is 
 really the case, and how much is thought to be the case because 
 of our traditions, it is impossible at present to say. It is a 
 fact, however, that the average school year is longer in Germany 
 than in America, and that our excessively hot weather in June 
 and September, not to speak of July and August, is practically 
 unknown there. Neither winter nor summer gives such extremes 
 of temperature as are found with us, and on the whole the 
 climate is better adapted to a long, steady pull. Whether we 
 make up for it by the energy and the rapid work of our stimu- 
 lating winters cannot be told. At any rate, as to the school 
 year, they have the advantage. 
 
 The same is true as to the school day. The Germans go to 
 school earlier in the morning and on the average they spend 
 more hours there than we do. They make up for the confinement 
 by long walks with their teachers on half holidays, and by 
 healthier play than is found in most American schools. The ele- 
 ment of time in contact with the teacher is therefore one with 
 which we must reckon. 
 
 Then, too, we are compelled to admit that their teachers are, 
 on the whole, better prepared than ours. They know their 
 subject better, and in their probation year they have been under 
 better guidance and more severe drill than we are giving in 
 this country. It is not to our discredit that this is so, for America 
 has done well when we consider our problem. Ours is a coun- 
 try where the natural resources have created great wealth within 
 a brief period. Young men have naturally and properly been 
 attracted into commercial fields rather than into the financially 
 unremunerative profession of teaching. Even if our nation had 
 grown in numbers through its own natural increase it would 
 have been difficult to furnish men for our high schools ; but with 
 our influx of a million immigrants a year we have been nearly 
 overwhelmed with the task of educating our youth. Had not 
 our women's colleges developed as they have in the past gen- 
 eration, we should have been in a sad state in our effort to secure
 
 73] The Present Teaching of Mathematics in Germany 7 
 
 teachers. When, therefore, we hear that our mathematics does 
 not measure up to the standard because of our great supply 
 of women teachers, we should remember that if it had not been 
 for the entry of women into the field of teaching we should 
 have had to face the dire disaster of illiteracy. 
 
 Then, too, it must be said that the leaders in educational matters 
 in Germany have more scholarship than has generally been the 
 case in America, and that they have naturally tended to appre- 
 ciate scholarship more highly. With us the attack on a subject 
 like the classics has generally been made by people who knew 
 little or nothing of Latin and Greek, and who have sought to 
 abolish what they did not understand, rather than to improve its 
 presentation. The same is true to some degree with mathematics, 
 and this accounts for some of our tendency to make the subject 
 merely one for the workshop. 
 
 Nevertheless the greatest defect in our system to-day, as 
 compared with Germany, is to be found in the lack of sound 
 mathematical training on the part of the teachers themselves. 
 Principles of education are valuable, psychology has come to 
 be a necessary part of a professional equipment, the history of 
 education must be studied to lead teachers away from the un- 
 fortunate attempts of the school anarchist, but underneath all 
 this must lie a solid foundation of mathematical knowledge if 
 the teacher is to lead students to know and to love the subject 
 and to know its bearings upon life. There is where Germany 
 has the advantage, and to the lessening of this advantage those 
 who desire the best for the American schools must address them- 
 selves rather than to an aimless iconoclasm. 
 
 A further advantage that the German school has is found in 
 its general plan. Our elementary school usually covers eight 
 years, the teaching being done by women who have rarely had 
 college training, and who, in spite of all their devotion to the 
 work, cannot have a broad mathematical outlook. The pupil 
 then enters the high school, and usually comes under the instruc- 
 tion of a teacher (and often again a woman) who has, at the 
 most, studied the calculus. The mathematical outlook is now 
 better, but it is rarely satisfactory. But there is a decided break 
 in the pupil's work after leaving the elementary school. In
 
 8 Teachers College Record [74 
 
 Germany there is no such break, for the Gymnasium extends 
 over nine years. For all this period the pupil lives in the same 
 general atmosphere, with the same corps of teachers. This is 
 the reason why Germany can successfully begin algebra and 
 geometry earlier than we can, and run them in unbroken sequence 
 for several years. If we begin algebra in the eighth grade 
 we do so with an instructor who is trained solely for elementary 
 work, and who has forgotten most of the algebra she once knew ; 
 but in Germany they can begin it in the seventh grade with the 
 same teacher who is afterwards to present the trigonometry, 
 one who knows mathematics thoroughly. Until we break away 
 from our idea of a four-year high school we cannot hope for 
 the " long pull and strong pull " of the German schools. 
 
 The reader will look in vain in the German reports for evi- 
 dence of the hysterical search after a plan for studying and 
 benefiting from mathematics without any work, a plan that seems 
 to be in evidence in some parts of this country. Germany recog- 
 nizes fully the varied capabilities of children, and of course the 
 Gymnasium is only one type (and a limited one) of school. 
 This recognition leads to a greater number of types than we have 
 developed, although we seem to be on the right road in building 
 up our various forms of industrial high schools. But nowhere 
 in Germany does there seem to have developed that kind of mind 
 that seems to seek, in this country, to abolish algebra and 
 geometry from a course of study designed to give an all-round 
 training. Vocational mathematics there is, but it is serious, and 
 in general it recognizes not alone the immediately practical but, 
 what is far more important, the potentially practical as well. 
 The tendency to pay the most attention to the less intellectual 
 type of mind is rather marked in America to-day. The best 
 thought should surely go as much to the encouragement of the 
 boy who wishes to develop intellectually as to the one who does 
 not wish to do so. In this country to-day one has to search not 
 a little to find a teachers' association that is discussing the 
 problem of teaching more mathematics to the boy who wants 
 to learn, but he has no trouble in hearing all sorts of plans 
 for teaching no mathematics at all to the one who seems ex-
 
 75] The Present Teaching of Mathematics in Germany 9 
 
 ceedingly anxious to learn nothing that makes for intellectual 
 advance and for higher ideals. 
 
 The contrast between German and American schools is not, 
 however, all to the advantage of Germany. It is by no means 
 certain that a smattering of analytics and the calculus would be 
 a good thing for our pupils, nor that it is a good thing for 
 theirs. While it would probably be a desirable plan to run our 
 algebra and geometry over a longer period if we had a six- 
 year high school with departmental teaching, and possibly side 
 by side (although there are other factors to be considered which 
 lack of space prevents mentioning in this chapter), still there is 
 an advantage in concentrating the attention upon a subject for 
 a briefer time. Our course in algebra is in many respects better 
 than the German course, and our geometry is quite as thorough 
 as theirs in the field covered, although nobody recognizes better 
 than we that there is room for improvement. With the present 
 four-year plan, with the present training of our teachers, and 
 particularly with the time now allotted to mathematics, to attempt 
 to cover more ground, save as we may offer electives in the few 
 schools that are prepared for such work, would be a decided 
 step backwards. The work that we are doing is by no means 
 bad work; we have exceptionally good courses in algebra and 
 geometry, and these we are constantly seeking to improve by 
 making them appeal more to the interests of the pupils, and by 
 eliminating material that we cannot justify. We usually offer 
 at least one year of elective work for pupils who are mentally 
 fitted to undertake it, and shall probably offer more in our best 
 schools in the near future. Our work is, therefore, far from 
 being all bad, nor need we be at all ashamed of what our better 
 schools are doing, or of what our system as a whole is accom- 
 plishing when all conditions are considered. The efforts put 
 forth to make mathematics interesting and vita! are apparently 
 quite as good in America as in Germany, and while we have a 
 great deal more to do in order to reach an ideal presentation of 
 these subjects, the spirit shown by our teachers in this respect 
 is excellent. Moreover we have a number of scientifically con- 
 structed text-books in arithmetic, algebra, and geometry, as good 
 indeed as can be found anywhere, considering our special needs.
 
 io Teachers College Record [76 
 
 These, again, must continually be improved, particularly in the 
 direction of furnishing a motive for the work for certain types 
 of mind, but this improvement is proceeding in a creditable 
 fashion. We have, therefore, not a little that stands to our 
 credit when we come to write up our accounts. So true is this 
 that it would be a foolhardy policy to attempt any wholesale 
 destruction of our present courses in algebra and geometry, or 
 any hasty modification of our curriculum. 
 
 The German reports should, however, convince us of several 
 things: 
 
 1. We need to bend every effort to give to our prospective 
 teachers a better knowledge of mathematics. 
 
 2. We then need to consider carefully the possibility of extend- 
 ing our present high-school mathematics over a longer period 
 with the same corps of teachers; in other words, to consider 
 the advantages of a six-year high school. 
 
 3. We also need to get ready to offer higher electives, including 
 trigonometry and its applications, geometric drawing (which we 
 are entirely neglecting), a form of practical (and potentially prac- 
 tical) mathematics that is not merely a little arithmetic, and 
 possibly enough of the calculus for simple work in mechanics, 
 and enough of analytics for appreciating this amount of the 
 calculus. These should be made worth the while for the stu- 
 dent who will never go to college, and they can surely be pre- 
 sented in such way as not to harm the one who is going to 
 pursue his studies further. 
 
 4. We also need to set our faces against mathematical work 
 that is scrappy and without scientific content. Work of this kind 
 is not found in such countries as Germany, countries that lead 
 in commerce, in agriculture, and in industry, as well in educa- 
 tional matters. Germany has not secured for mathematics the 
 status that it has in all her schools by yielding to the demands 
 for weak algebra and weak geometry, or for none at all, on the 
 part of men who know nothing of the subject save what they 
 got from a poor course in some high school. Nor shall we 
 make a worthy place for the subject in our curriculum by con- 
 sidering only the weakest minds, only the immediately practical, 
 and only that which any untrained teacher can present.
 
 77] The Present Teaching of Mathematics in Germany n 
 
 5. With respect to vocational training we must recognize that 
 a new type of mind has appeared in our high schools, demanding 
 a new type of mathematics. But that this must be weak mathe- 
 matics, or only the immediately usable, is not the experience of 
 the nation that has made the greatest strides of all in industrial 
 work in the past generation. We may well take counsel of that 
 nation in training our future artisans in the power of thinking 
 clearly and logically along mathematical lines. 
 
 6. And finally, there is great necessity for continued advance 
 in the training of leaders among the teachers of our country. 
 The average college cannot do this work under present condi- 
 tions. Indeed the feeling is still prevalent in many colleges that 
 professional training is unnecessary. One of the most important 
 lessons that Germany teaches us is the fallacy of this position. 
 The establishing of strong courses for teachers in the senior 
 year of our colleges will do much to improve our educational 
 work, particularly in our secondary schools.
 
 CHAPTER II 
 EVOLUTION OF THE REFORM IN GERMANY 1 
 
 Isidore Skolnick 
 
 The report by Dr. Schimmack, referred to in the footnote, is 
 divided into two sections ; the first dealing with the reform and 
 its progress from 1840 to 1907, and the second dealing with the 
 reform and its progress from 1907 to the present day. 
 
 Professor Felix Klein in the introduction to this report refers 
 to a statement made by himself at Gottingen in March, 1911, 
 that for some time he had wished to bring before the public the 
 discussion of the Commission, and to make known his own per- 
 sonal opinions and his general aim. What he means to carry 
 out to-day is a plan which he has heretofore been unable to 
 introduce. The idea foremost in the minds of many teachers 
 of mathematics in Germany at present is that the concept of 
 " function " as defined in mathematical language should be the 
 central core around which the student's knowledge of mathe- 
 matics is to be built. The purpose in introducing the idea of 
 function is not to teach a barren analytic geometry but to make 
 clear the simple and all-important notion that upon the changing 
 of x the changing of y depends. It is expressly stated that the 
 function idea should be continually borne in mind in the teaching 
 of algebra and geometry. It is urged that no abstract function 
 idea is to be taught, but, on the contrary, the concrete relations 
 existing between x and y as functions of each other are to be 
 expounded. 
 
 It may be of interest to summarize briefly the first part of 
 the report, which is a general survey of the teaching of mathe- 
 matics from the year 1840 to 1860. During this period neither 
 
 1 Die Entwicklung der Mathematischen Unterrichts Reform in Deutsch- 
 land. von Dr. Rud. Schimmack, Oberlehrer am Gymnasium zu Gottingen, 
 Leipzig und Berlin. 191 1. 
 
 12 [78
 
 79] The Present Teaching of Mathematics in Germany 13 
 
 the idea of function nor the graph was presented in the schools 
 of Germany. Dr. Schimmack remarks that it seems strange 
 that anyone could understand analytic geometry, maxima and 
 minima, or the calculus, without being familiar with this func- 
 tion idea. 
 
 In order that this reform movement may be traced more defi- 
 nitely and closely, it will be well to follow the various changes 
 that have taken place from time to time in Prussia. As early 
 as 1812, in one of the Prussian Gymnasien, the following courses 
 in mathematics were given: a very few simple computations in 
 logarithms; the elementary geometry of Euclid from book one 
 through book six, together with books eleven and twelve; and 
 a little plane trigonometry. Later on, about 1834, progressions 
 and permutations and combinations were added. In 1835 spher- 
 ical trigonometry and a little work in conies were also found in 
 the course. 
 
 The Lehrplan for the Prussian Gymnasium from 1837 to 1856 
 shows that in general there was no change from that of 1835. 
 Up to 1867 no analytic geometry, maxima and minima, or the 
 calculus were to be found in the Lehrplan. The only additional 
 subjects given for the Realschulen of the first order (those of 
 1867 which later became the Realgymnasia) were the following: 
 elements of descriptive geometry, a little work in conies, and 
 also some statics and mechanics. The instructor, if he were 
 capable, tried to supply some further work in analytics, some 
 differential and integral calculus, and a little astronomy. All 
 of the above was given in the schools, of Prussia, Schleswig- 
 Holstein, Hanover, and Hesse-Nassau. Outside of Prussia the 
 Lehrplan was no better except that here and there the idea of 
 function was beginning to creep in. On the whole, much less 
 mathematics was to be found outside of Prussia than in Prussia 
 itself. 
 
 About the year 1859, K. H. Schnellbach, a member and organ- 
 izer of the Examiners' Commission, founded and conducted a 
 mathematical pedagogical seminar. About the same time a book 
 appeared on the methods of mathematical teaching, by Mekler 
 (Das Mekler Lehrbuch, 1859). This book contains problems in 
 planimetry, stereometry, and trigonometry, with some work in
 
 14 Teachers College Record [So 
 
 physics involving functions. The author gives a brief discussion 
 of simple curves related to rectangular co-ordinates. He makes 
 use of the function in maxima and minima, of graphs, and 
 of the elements of the calculus. Another noteworthy book, 
 Baltzer's " Elemente der Mathematik," followed, and was in 
 great demand in the teachers' seminar. This book dealt with 
 algebra, the new geometry, analytics, and the calculus. The 
 period from the year 1870 to 1890 is characterized by a number 
 of important changes. For example, a Schulkonference met in 
 1873 and recommended that analytic geometry as well as the cal- 
 culus be considered as courses in the schools. 
 
 From the Lehrplan of the Prussian Gymnasium of 1882 it 
 would appear that mathematical teaching was neither advancing 
 nor going backward. It was urged by those who framed it that as 
 much mathematics should be taken up as would render one 
 able to understand the mathematics of geography, to deal with 
 conies intelligently, and, as far as possible, with differential 
 quotients. In this Lehrplan very little of analytics and spherical 
 trigonometry was included. 
 
 In 1884, Professor M. Simon, Oberlehrer of the Lyzeurh Gym- 
 nasium at Strassburg, advanced the proposition that the ele- 
 ments of algebra should be a preparation for the theory of func- 
 tions. About the same time Dr. A. Hofler of Vienna suggested 
 that it would be well if Cartesian co-ordinates were taught in 
 Class VII of the Gymnasium. To show more clearly what 
 courses he urged, he gave certain curves that might be plotted 
 in this grade: 
 
 y 2 = ex, x 2 y 2 = c, ax 2 by 2 = c, 
 xy = a, y = ax', x^" = a. 
 
 For transcendental curves he suggested y = log x and y = sin x. 
 It was his idea that the plotting of these curves would give a 
 more concrete meaning to the function. For the same purpose 
 he urged the presentation of the temperature curve and its 
 application to physics. Such curves as those of mortality were 
 also suggested. He asserted that the plotting of the above curves 
 might easily be introduced in the first three classes of the Gym- 
 nasium. 
 The years from 1890 to 1910 are marked by even more radical
 
 81] The Present Teaching of Mathematics in Germany 15 
 
 changes. About the year 1890 the question arose as to whether 
 or not conies should be taught in the Gymnasium, and it was 
 discussed in a gathering .of teachers of mathematics in January 
 of that year, with the result that it was decided to encourage 
 the teaching of analytics in the Gymnasium. 
 
 In the Prussian curriculum of 1892 it was agreed that in the 
 last classes of the Gymnasium there should be a more detailed 
 study of rectangular co-ordinates, with special reference to conies. 
 It was also arranged that analytic geometry in general should be 
 a required subject in the course of study. Outside of Prussia, 
 the students of the Gymriasien seem to have had no knowledge 
 of co-ordinates or the function, nor does anything seem to have 
 been done with analytic or advanced synthetic geometry. 
 
 In the Lehrplan of 1901, the function concept was given a 
 prominent place and the connection between physics and mathe- 
 matics was emphasized. Dr. E. Catling, Oberlehrer of the Gym- 
 nasium at Gottingen, was among those who expressed very posi- 
 tive opinions that the variable and the function should be used 
 and taught in algebra, geometry, trigonometry, and analytics. 
 ' Through ignorance of the function idea," he asserted, " one 
 makes mathematics hard for himself." 
 
 It was in 1904 that Professor Klein made this important state- 
 ment : " The geometric function idea will stand as the important 
 or central point of mathematical teaching and will be carried 
 as far as possible, and by this means the elements of the dif- 
 ferential and integral calculus will be introduced into the high 
 schools of Germany." The influence of this remark has been 
 very far reaching. 
 
 The reforms since 1907 have been most important and are 
 discussed in detail in the second part of the report. At the 
 beginning of that year considerable discussion arose as to 
 the relation between mathematics and the sciences, and conse- 
 quently a commission was appointed for the purpose of suggest- 
 ing various courses in these two lines of work. This commission 
 decided upon making the courses in mathematics and sciences 
 distinct, and accordingly a separation between these subjects was 
 agreed upon for the high school. One group was called the 
 physics-mathematics group and the other the chemistry-biolog-
 
 16 Teachers College Record [82 
 
 ical group. The idea of this separation was to give more de- 
 tailed courses in physics, chemistry, biology, and mathematics. 
 This commission was called the Commission of German Natural- 
 ists and Physicians, and it met at Dresden in September, 1907. 
 It also met at Leipzig in 1908, and at this time discussed the 
 subject more critically. 
 
 Another commission met in 1908 to decide what should be 
 done with regard to the sciences and mathematics in the Girls' 
 High School. Dr. A. Gutzmer drew up a report upon the 
 details of its discussions. A little later another report appeared 
 dealing with the question of the place of mathematics and sciences 
 in the new girls' high schools of Prussia. At one of the meet- 
 ings of the commission. Dr. E. Hoppe, of Hamburg, told the 
 assembly that the teaching of mathematics and sciences according 
 to the reform plan was proving to be a great success. He said 
 that the idea of the function was given to the students in the 
 Obersekunda, and that they found it interesting and very easy 
 to understand. He stated to the committee that this led him to 
 extend the function idea in the humanistic Gymnasium. Dr. 
 H. Schotten, a successful teacher of the reformed mathematics 
 in this high school, also showed how the work could be presented. 
 
 The remainder of the report by Dr. Schimmack is devoted to 
 a discussion of the work of the International Commission and 
 the reports of the various sub-committees, all of which are 
 reviewed in detail in the following chapters. It will perhaps be 
 helpful to mention one or two facts which are not brought out 
 in the separate reports. 
 
 In tracing the growth of the Gymnasium in Prussia from 1901 
 to 1909, we notice a gradual increase in the number of students 
 from 87,478 to 102,297. In the Prussian Realgymnasium there 
 is an increase from 21,078 to 41,202. In the Oberrealschule the 
 increase has been from 14,800 to 34,735 during the same time. 
 These statistics show that the increase has been greatest in the 
 Oberrealschulen. In Bavaria alone we find nine such schools, 
 namely : Augsburg, Passau, Reglaburg, Bayreuth, Kaiserslantern, 
 Ludwigshofen (Rhein), Munich. Number^, and Wiirzburg. 
 These schools give courses in general educational subjects similar 
 to those given in the Gymnasium and the Realgymnasium of
 
 83] The Present Teaching of Mathematics in Germany 17 
 
 Prussia. The following table gives the number of hours devoted 
 to mathematics in the Oberrealschulen of Prussia, Bavaria, Sax- 
 ony, and Baden: 
 
 LOWER UPPER LOWER UPPER LOWER UPPER 
 Classes 6 543 3 2 2 1 1 
 
 f Arith. 5 
 
 PRUSSIA \ Math. -566 5 5 5 5 5 
 
 I Math, draw'g (2) (2) (2) (2) (2) 
 
 f Arith. 4431 - ___ 
 
 BAVARIA \ Math. 24 5 5 5 5 5 
 
 [Math, draw'g --1 1 (2+) (2+) 
 
 f Arith. 4442 2 1 
 
 SAXONY j Math. --24 4 5 6 6 6 
 
 [ Math, draw'g - 1 222 
 
 f Arith. 5 _ _ _ __ __ 
 
 t, Ar \ Math. -555 5 5 5 5 5 
 
 BADEN I Math, draw'g - 2222 
 
 This shows that mathematical drawing is compulsory in Sax- 
 ony, Baden, and Bavaria, but is not compulsory in Prussia. 
 
 In 1905 a new Lehrplan was drawn up to meet the more recent 
 demands for the girls' high school. In 1908 a new school 
 was established which admitted girls at the age of six. After 
 completing this course they could enter the Lyzeum or Frauen- 
 schule, the latter containing also a teachers' seminar for women. 
 In this high school, mathematics is an important subject and is 
 given three hours a week throughout the course. In Prussia 
 and Saxony the course followed is that indicated by the Prussian 
 Lehrplan of 1908. This is a ten-year course which the girls 
 begin at the age of six. 
 
 Dr. Schimmack's report also gives a list of books and their 
 titles for the use of the secondary schools. Among the books 
 mentioned are Behrenden and Gotting's " Lehrbuch," P. Crantz's 
 " Book for Girls' High Schools," and H. Dressler's " Study of 
 the Function." There is also given a list of books for the use 
 of the instructor. This includes such books as E. Borel's " Ele- 
 ments of Mathematics," J. Druxes' " Book on Arithmetic and 
 Mathematics," F. Klein's " Mathematical Teaching in High 
 Schools," and " Elementary Mathematics for the High School," 
 and O. Lesser's " Graphs in the Mathematics for High Schools."
 
 CHAPTER III 
 SECONDARY SCHOOLS OF HESSE AND BADEN 
 
 Miriam E. West 
 
 The two reports that are reviewed in this chapter 1 treat of the 
 organization, the curriculum, and the methods of mathematical 
 instruction in the higher schools of Hesse and Baden. The 
 schools of Hesse and Baden follow in general the plan of or- 
 ganization of the higher schools throughout Germany, the one 
 exception being that in Baden three and one-half instead of 
 three years are required in preparation for entrance. 
 
 HESSE 
 
 In the Grandduchy of Hesse, a country with a population of 
 about one hundred and twenty-six thousands, there are the fol- 
 lowing schools: ii Gymnasien, 3 Realgymnasien, 7 Oberreal- 
 schulen, 9 Realschulen, 5 higher girls' schools, and 33 higher 
 Burgerschulen. These last, which are for the most part five-class 
 schools of the type of the Realschule, are found in small places. 
 A few have a complete seven-year course. There are several 
 of these exclusively for girls. The higher girls' schools are ten- 
 class schools admitting girls at six years of age. In connection 
 Vvith some of these there is an additional course of three or four 
 years for the training of teachers. 
 
 In the lowest classes of the girls' schools the fundamental 
 operations are taught, and are followed in the fifth year by 
 
 1 Der Mathematische Unterricht an den Hoheren Schulen nach Organisa- 
 tion, Lehrstoff und Lehrverfahren und die Ausbildung der Lehramtskandi- 
 daten im Grossherzogtum Hessen, von Professor Dr. Heinrich Schnell, 
 Oberlehrer am Ludwig-Georgs-Gymnasium in Darmstadt. April, 1910. 
 _ Der Mathematische Unterricht an den Hoheren Schulen nach Organisa- 
 tion, und Lehrverfahren und die Ausbildung der Lehramtskandidaten im 
 Grossherzogtum Baden, von Hans Cramer, Professor an der Goethe- 
 schulc in Karlsruhe. July, 1910. 
 
 18 [84
 
 85] The Present Teaching of Mathematics in Germany 19 
 
 commercial measures and the subject of fractions, which is 
 completed in the sixth year. In the last four years business arith- 
 metic is studied, and in the ninth and tenth years some geometry. 
 In the ninth year the fundamental ideas of geometry and the 
 computation of plane figures are taken up, and in the tenth year 
 the computation of volumes and surfaces of solids. In the 
 teachers' training course this work is reviewed and extended 
 with especial emphasis upon rigorous proof. 
 
 Of the three types of schools, Gymnasium, Realgymnasium, 
 and Oberrealschule, the Oberrealschule devotes the greatest num- 
 ber of hours to mathematics and the Gymnasium the least. The 
 nine classes in these three schools are numbered, beginning with 
 the lowest class : VI, V, IV, Illb, Ilia, lib, Ha, Ib, la. One 
 year is required for the completion of each class. The topics 
 treated in the various classes of the Realgymnasium are as 
 follows : 
 
 HRS. 
 CLASS PER 
 
 WEEK 
 
 SUBJECT 
 
 VI 6 ARITHMETIC: review of fundamental operations with whole 
 numbers, abstract and denominate; addition and subtrac- 
 tion of decimal numbers; factoring. 
 
 V 4 ARITHMETIC: common fractions with concrete illustrations; 
 decimal fractions. 
 
 IV 5 ARITHMETIC: common and decimal fractions; business arith- 
 metic; simple rule of three. 
 
 Plane geometry through the congruence propositions for 
 triangles; practice in use of ruler, compasses, squares, etc. 
 
 Ill b 5 ARITHMETIC: business arithmetic continued. 
 
 Plane geometry: systematic instruction begun. 
 
 Ilia 
 
 II a 
 
 ALGEBRA: through powers and roots. 
 Plane geometry continued. 
 
 II b 5 ALGEBRA: logarithms; equations of the first degree; quad- 
 ratic equations. 
 
 Geometry: computation of the circle; introduction to plane 
 trigonometry. 
 
 ALGEBRA: quadratic equations with more than one unknown; 
 interest and annuities.
 
 20 
 
 Teachers College Record [86 
 
 HRS. 
 CLASS PER SUBJECT 
 
 WEEK 
 
 I b 5 TRIGONOMETRY continued; stereometry. 
 
 ALGEBRA: diophantine equations; combinations; figurate 
 numbers; arithmetic series of higher order; binomial 
 theorem for positive integral exponents; De Moivre's 
 theorem; cubic equations. 
 
 STEREOMETRY continued ; elements of spherical trigonometry. 
 
 la 5 ALGEBRA: determinants; transcendental series. 
 ANALYTIC GEOMETRY of straight lines and conies. 
 
 The Oberrealschule has in addition to these a course in 
 geometric-drawing in which the following topics are considered : 
 
 lib and Ha CONSTRUCTION of plane figures and oblique geometrical solids. 
 
 Ib DESCRIPTIVE GEOMETRY: review and extension of the funda- 
 mental propositions; projection of solids; plane sections 
 of solids. 
 
 la REVIEW : solution of difficult fundamental problems with 
 representation in oblique projection; shadow construction; 
 elements of perspective; solution of simple practical 
 problems; right-angled axonometry and oblique parallel 
 projection. 
 
 The arithmetic as given offers several opportunities for an 
 introduction to algebra. In the review of the fundamental opera- 
 tions in the first year the pupil is made familiar with parentheses. 
 Fractions are considered with a special view to that topic in 
 algebra. The rules given are such that they are applicable in 
 algebra as well as in arithmetic. In business arithmetic, by the 
 use of formulas to express the rules, the literal symbols of 
 algebra are introduced. 
 
 The instruction in algebra begins with the concept of the gen- 
 eral number for which the way has been prepared in arithmetic, 
 if the mathematical instruction is well organized. The instruc- 
 tion in proportion is correlated with that in geometry. There is 
 a tendency in the reform movement to break away from the 
 old Euclidean type of proportion, which has its only value in 
 geometry. The equating of the products of the terms will 
 answer every purpose in algebra. Another reform has been 
 to lessen the extent to which formalism rules the subject matter 
 in algebra. There are many illustrations of this formalism: 
 for instance, the division of long polynomials, a process which
 
 87] The Present Teaching of Mathematics in Germany 21 
 
 is little used in the later work; in the subject of powers and 
 roots, the solution of problems involving high degrees and com- 
 plicated forms which, likewise, are of little value in later work. 
 The problems in equations as well as many of those in interest 
 and annuities are not practical. In many schools such topics 
 as cube root, arithmetic progressions of higher order, cubic equa- 
 tions, combinations, and diophantine equations are omitted. 
 
 At present little use is made of the graphic representation of 
 functions. In the schools which have adopted the reform plan 
 the graphic representation of the linear function, and of y = x 2 
 and y = x 3 , is introduced. Applications are found in physics and 
 in the solution of equations. The graphic solution of the quad- 
 ratic of one unknown quantity is not practical. By the use of 
 the graphic solution of the cubic equation one is spared the 
 complicated algebraic method, but with quadratic equations the 
 graphic method is not easier than the algebraic. The graphic 
 treatment is of value, however, in quadratic equations of two 
 unknown quantities in showing the existence of the different 
 roots. Its use here furnishes an excellent introduction to analytic 
 geometry. 
 
 Plane trigonometry follows the course in geometry. The func- 
 tions of the angles and computation of right triangles are con- 
 sidered. Logarithms find application here, but many teachers 
 prefer to use the natural value of the function first. To meet 
 this desire the following plan has been devised. 
 
 lib Trigonometry limited to sine and cosine proposition. Stereometry 
 
 limited to the consideration of solids. 
 Ha General stereometry, goniometry, and logarithms. 
 
 Where this plan is adopted those who leave school at the close 
 of lib will have no unnecessary logarithms but will have stereom- 
 etry. Spherical trigonometry, which is taken up in the last year, 
 is applied to mathematical geography. 
 
 Analytic geometry forms the conclusion of the geometrical 
 course. In it the study of both branches of school mathematics, 
 algebra and geometry, is brought to a close. Some teachers use 
 the analytical method of treatment and some use the geometrical 
 method, while many combine the two as in the following plan. 
 The sections of the cone and the focal properties of the curves 
 are considered in stereometry. This serves to introduce the
 
 22 Teachers College Record [88 
 
 analytical treatment, in which problems formerly handled graph- 
 ically are turned around by asking the quertion, whether it is 
 possible to find the functions of the different curves, having given 
 their graphic representation. In this manner the different 
 branches of mathematical instruction are brought together anew. 
 In the realistic schools the pupils are able at the end of the course 
 to find the equations for simple geometrical loci. 
 
 The consideration of curves of all kinds leads naturally to 
 the idea of differential quotients, and the theory of maxima and 
 minima. Thus in nearly all of the realistic schools some work 
 is done in differential calculus. The teachers in the Gymnasien, 
 however, object to the use of the symbols, although they do not 
 object to the introduction of the ideas of calculus. With the 
 exception of the study of the history of mathematics in the Ober- 
 realschule at Heppenheim, this completes the instruction in 
 mathematics given in the higher schools of Hesse. 
 
 BADEN 
 
 Besides the three types of nine-class schools in Baden there 
 are a few six- and seven-class schools. The general manage- 
 ment of the various institutions is vested in the state, with the 
 exception of the Realschulen and the girls' schools, which are 
 partly supported by the community. Certain privileges therefore 
 concerning selection of teachers and management are granted to 
 the community. For entrance to any of these schools the child 
 must pass an examination. Girls attend the higher boys' schools 
 where there are no institutions especially for them. In 1909, 
 8.2 per cent of those attending these schools were girls. At first 
 only the Biirgerschulen were open to them, but since 1900 they 
 have been admitted to the other schools. The greatest number 
 are found in the realistic type of school. 
 
 Some of the distinguishing features of this report are as 
 follows. The amount of mathematics offered in the schools 
 of Baden does not differ greatly from that outlined above for 
 those in Hesse. In the girls' schools there is more work done 
 in mathematics than in those of Hesse. The work in algebra 
 includes equations of the first degree with one and two unknown 
 quantities, powers, roots, and proportion.
 
 89] The Present Teaching of Mathematics in Germany 23 
 
 In the boys' schools, aside from slight differences in the 
 arrangement of topics, there are three ways in which the instruc- 
 tion differs from that in Hesse: (i) The intuitive instruction 
 given in geometry in the Oberrealschulen distinguishes its plan 
 of instruction from that of the other German states. (2) Ac- 
 cording to the report the use of models is much more ex- 
 tensive. (3) The elements of differential and integral calculus 
 are found in a large number of schools of all three types. 
 
 The intuitive instruction in geometry is begun in class V and 
 continued through three years, followed in class Ilia by the 
 formal instruction. In the first year, knowledge is gained con- 
 cerning the various plane figures and their properties by means 
 of looking at the solids and plane figures as well as by drawing 
 and construction. In the second year some of the facts con- 
 cerning the equality of plane figures are observed and areas 
 are computed. In the third year the instruction concerning 
 solid figures proceeds in a similar manner. If this instruction 
 is properly carried out it is not a mere diluted form of the 
 scientific instruction, nor is there a line sharply dividing the 
 two. First the pupil is asked to state the results of his observa- 
 tion, and after several observations he arrives at a general state- 
 ment of the facts. The next step is to arrive at facts that are 
 not so evident, but must be derived by the aid of those he 
 already has. So the pupil gradually comes to feel the need of a 
 proof. It is a method which makes use of the eye, hand, and 
 mind of the pupil. 
 
 The following is a list of some of the models and apparatus 
 used : for elementary arithmetic, dry measures of commercial 
 forms, cubic decimeter and centimeter of wood or metal, spheres 
 and circular plates in whole or in parts for fractions; for ele- 
 mentary geometry, wooden models of all geometric forms, a 
 large number of models for volumes and areas, made out of 
 wood and jointed ; for practical trigonometry, >all the various 
 surveying instruments. The recommendation includes models 
 for various propositions in stereometry and spherical trigonom- 
 etry; sections of cones and cylinders, the cutting plane being of 
 different material ; spheres with zones and sectors cut out ; models 
 especially designed for drawing; and the slide rule for the use 
 of the pupils.
 
 24 Teachers College Record [90 
 
 The course in the infinitesimal calculus includes the following : 
 maxima and minima, formation of infinite reries, determination 
 of lengths of arcs, surface area, and volume of simple solids of 
 revolution. Most teachers agree that the concept of the function 
 and the graphic representation of it should be introduced as 
 early as class III, but the differential quotients and integration 
 should be left for the last two years. Problems in calculus occur 
 in the final examinations of many schools. 
 
 The final examination in mathematics, which is given at the 
 completion of the nine-years' course, is both oral and written. 
 The written part consists of four problems, two from algebra and 
 two from geometry. These are selected by the councilor of the 
 state board from problems made out by the mathematics teacher 
 of the upper class of the institution where the examination is 
 to be held. The problems are closely related to the mathematics 
 of the last school year. Besides this the pupil has to write a 
 theme, the subject of which is chosen from topics considered 
 in the upper class. 
 
 In reading these two reports we are impressed with the unity 
 in the mathematics curriculum, a fact which may not stand out 
 so prominently in this brief review. This unity is accomplished, 
 not by a general mixing of the different branches of mathematics, 
 but by allowing each branch to serve as a natural introduction 
 to that which follows. As stated before, the work in formulas 
 in arithmetic prepares the way for the literal symbolism of alge- 
 bra. The graphic work in algebra and the study of conies in 
 stereometry serve as an introduction to analytic geometry. As 
 soon as the necessary foundation for trigonometry has been laid 
 in geometry, that subject is begun. Logarithms are taken up in 
 algebra at the time when they are needed for trigonometry. In 
 Baden the preparation for the study of calculus is begun early 
 in the course by the introduction of the concept of the function 
 and its graphical representation. The fact that the instruction 
 in mathematics is under the direction of special mathematics 
 teachers for at least six years makes this unity to a large degree 
 possible. We may well afford to compare this with the isolated 
 teaching of the different branches of mathematics in the United 
 States, to see if we cannot learn a lesson from Germany.
 
 CHAPTER IV 
 
 THE SECONDARY SCHOOLS OF THE HANSEATIC 
 
 STATES 
 
 {Catherine S. Arnold and Ruth Fitch Cole 
 
 The report on mathematical instruction in the Gymnasien and 
 Realschulen in the Hanseatic states, Mecklenburg and Olden- 
 burg, 1 was prepared from the results of investigations carried on 
 by sending questionnaires to the various educational centers. The 
 data gathered in this manner gave information concerning the 
 curriculum, the methods of presentation of mathematical material, 
 and the topics emphasized in the teaching of the various subjects. 
 Although the opinions received from the various instructors in 
 mathematics disagreed in a few details, they showed certain 
 general tendencies, which are embodied in this report. 
 
 The course of study in a Gymnasium, a Realgymnasium, and 
 an Oberrealschule, the three types of secondary schools in Ger- 
 many, is given in order to show what topics are included in sec- 
 ondary instruction in mathematics. A plan from a reform Gym- 
 nasium is also added for the purpose of indicating the influence 
 of the new movement, the so-called reform movement, led by 
 Professor F. Klein. 
 
 All secondary schools have a nine years' course, three years 
 being required for entrance, so that the first year of a Gymnasium 
 or Real institution corresponds to the fourth grade of our Amer- 
 ican schools. 
 
 The following table gives the number of hours per week de- 
 voted to mathematics in the various classes of the three types 
 of schools in Hamburg, which is taken as a representative city. 
 
 1 Der Mathematische Unterricht an den Gymnasien und Realanstalten 
 der Hansestadte, Mecklenburgs und Oldenburgs, von A. Thaer [Ham- 
 burg], N. Geuther [Giistrow], A. Bottger [Oldenburg], Leipzig and 
 Berlin, 1911. 
 
 91] 25
 
 26 
 
 Teachers College Record 
 
 [92 
 
 Class 1 
 
 Gymnasium 
 
 Realgymnasium 
 
 Oberrealschule 
 
 VI V IV UIII OIII UII OH UI 01 Total 
 
 4 33 
 
 5 38 
 5 46 
 
 For convenience of comparison, the original Gymnasium course 
 and the suggested reform course are given in parallel columns. 
 This work begins in the upper third class, which corresponds to 
 our seventh grade. 
 
 GYMNASIUM 
 
 U 
 Algebra, through linear equations 
 
 with one unknown. 
 Geometry, single numerical exer- 
 cises with plane figures. 
 
 Equations with more than one un- 
 known, powers and roots. 
 
 The circle, measurement of rectilinear 
 figures, similarity, constructions. 
 
 .REFORM PLAN OF WILHELM 
 
 GYMNASIUM 
 Ill- 
 Algebra, linear equations and fac- 
 toring. 
 Geometry, the circle. 
 
 Ill 
 
 Equations with more than one un- 
 known, geometric construction of 
 irrational quadratic roots. 
 
 Regular polygons, proportionality. 
 
 U II 
 
 Quadratics, irrational quantities, 
 logarithms, exponential equations, 
 graphs of linear and quadratic 
 functions. 
 
 Measurement of polygons and circles. 
 
 Equations of the second degree with 
 one unknown, powers and roots, 
 logarithms. 
 
 Proportionality of 
 monic division, 
 the circle. 
 
 the circle, bar- 
 measurement of 
 
 O II 
 
 Powers with fractional and negative 
 exponents, progressions, interest, 
 and income. 
 
 Algebraic geometry, goniometry, trig- 
 onometry, beginnings of stereo- 
 metry. 
 
 Imaginaries, exponential equations, 
 systematic treatment of the func- 
 tion concept. 
 
 Extension of geometric representa- 
 tion of algebraic expressions, trig- 
 onometry. 
 
 U I 
 
 Plane and spherical trigonometry, go- 
 niometry, stereometry, analytic 
 geometry of the point, line and 
 circle. 
 
 Elements of spherical trigonometry, 
 arithmetical and geometrical series 
 of the first order, cubic equations, 
 stereometry. 
 
 O I 
 
 Coordinates, analytic geometry of the 
 conic sections, synthesis, maxima 
 and minima, map projections, re- 
 view. 
 
 Combinations, probability, binomial 
 theorem, analytic geometry, ele- 
 ments of differential and integral 
 calculus. 
 
 | The lowest class is the sixth, then the fifth, fourth, lower third, upper 
 third, etcetera.
 
 93] The Present Teaching of Mathematics in Germany 27 
 
 Below is given the course in mathematics for two types of 
 secondary schools. 
 
 REALGYMNASIUM OBERREALSCHULE 
 
 U Ill- 
 
 Algebra to linear equations with one 
 
 unknown. 
 The circle, constructions. 
 
 Algebra to linear equations with one 
 
 unknown. 
 Similarity of plane figures. 
 
 O Ill- 
 
 Proportion, powers and roots, equa- 
 tions of first degree with more than 
 one unknown, pure quadratics. 
 
 Similarity of plane figures, measure- 
 ment of rectilinear figures, construc- 
 tions. 
 
 Proportion, powers and roots, equa- 
 tions of first degree with more than 
 one unknown, pure quadratics. 
 
 Proportionality of straight lines and 
 circles, polygons, circumference and 
 area of the circle. 
 
 U II- 
 
 Quadratic equations, fractional and 
 negative exponents, logarithms. 
 
 Review of plane geometry, algebraic 
 geometry, projections, elements of 
 trigonometry. 
 
 O 
 
 Progressions, exponential equations, 
 quadratic equations. 
 
 Harmonic points, rays, poles, simi- 
 larity of points and axes. 
 
 Goniometry, trigonometry with field- 
 work, stereometry continued, de- 
 scriptive geometry, spherical trig- 
 onometry and applications. 
 
 Quadratic equations, arithmetical and 
 geometrical series of first order, 
 logarithms. 
 
 Fundamentals of goniometry, stere- 
 ometry, elements of trigonometry. 
 
 II 
 
 Trigonometry and stereometry con- 
 tinued, spherical trigonometry with 
 applications, quadratics. 
 
 L 
 
 Complex numbers and their geometri- 
 cal representation, cubic equations, 
 synthetic treatment of conic sec- 
 tions, map projections, applications 
 of spherical trigonometry to land, 
 nautical, and astronomical compu- 
 tations. 
 
 U I 
 
 Applications of algebra to geometry. 
 
 Analytic geometry of the plane. 
 
 Complex numbers, combinations, bi- 
 nomial, theorem, cubic equations, 
 insurance. 
 
 O I 
 
 Combinations, probability, binomial 
 theorem, differential calculus, max- 
 ima and minima, indeterminate 
 forms, infinite series, analytic geom- 
 etry of the plane, central projec- 
 tion. 
 
 Differential calculus, maxima and 
 minima, notion of continuity, inde- 
 terminate forms, curves, elements 
 of integral calculus with applica- 
 tions to geometry and physics. 
 
 Foremost among the questions which have occupied the atten- 
 tion of the commission is that of the best place in the course of 
 study in which to introduce the graphical representation of 
 functions. There are three opinions in regard to the matter:
 
 28 Teachers College Record [94 
 
 First, that the fifth year is not too early for the elementary 
 notions of functions; second, that it is in the beginning of 
 trigonometry that the subject comes most naturally; third, that 
 it is in physics that the most advantageous opportunity for its 
 introduction is offered. 
 
 Another point which has been considered is the subject of 
 drawing in connection with geometry. Accurate constructive 
 drawing is considered necessary to the satisfactory teaching of 
 mathematics. In some schools, neither the teachers nor the pupils 
 can draw properly. The difficulty is that the teachers are not 
 trained along this line, and it is evident that no assistance can 
 be given by the teachers of drawing. The only ones who can 
 improve the situation are the mathematics teachers themselves, 
 who should be skilled in drawing. In connection with courses 
 in stereometry, in several schools excellent drawing is done, and 
 in one Gymnasium working drawings are constructed. 
 
 The instruction in geometry has also been found unsatisfactory 
 in the initial presentation of the subject to the pupil. Among 
 the suggestions that have been offered are the following: first, 
 that a pre-geometric course be introduced leading up to the usual 
 theorems and exercises; second, that there should be a correla- 
 tion of like propositions in plane and solid geometry (this plan 
 originated in Italy, and while it was adopted in many schools 
 in Germany, it is no longer considered with favor) ; third, that 
 geometry should be approached from the standpoint of its 
 practical applications ; and fourth, that the approach on the 
 theoretical side is preferable. 
 
 As a further means of vitalizing mathematical instruction, a 
 brief historical sketch in connection with the several topics has 
 been suggested, it being argued that methods of discovery of 
 the fundamental principles of mathematics are not only interest- 
 ing but necessary for the student to know, not only giving him 
 a broader outlook but also suggesting the manner of approach 
 for future original research. When general history is presented, 
 why not associate each important event with its mathematical 
 contemporary, for instance, Hannibal and Archimedes, the 
 Reformation and the cubic equation, the Thirty Years' War and 
 analytic geometry, Napoleon and protective geometry?
 
 95] The Present Teaching of Mathematics in Germany 29 
 
 Opinion varies as to the manner in which the text-book in 
 mathematics should be used. While some maintain that the book 
 should be for the use of the teacher only, and thus of too high 
 a standard for the capabilities of the pupil, many more consider 
 that a suitable book should be placed in the hands of the pupil. 
 Some express themselves as desirous of a manual which shall 
 serve as a mere outline, while others think that the book should 
 contain a detailed explanation of the work. In general, there 
 seems to be a growing opinion that the pupil should have access 
 to a good book which contains the demonstrations and which 
 can be used for a model. Without such a guide, it is difficult 
 for him to construct a concise, logical proof of his own. 
 
 There is considerable discussion on the question of what 
 apparatus is to be used in the teaching of mathematics : whether 
 or not models shall be in the hands of the pupil, and if so, shall 
 these be constructed by him. There are times when a model 
 is a necessity, for example, in beginning the teaching of stereom- 
 etry; and the blackboard sphere is very desirable in gaining 
 the concepts of spherical trigonometry. However, the excessive 
 use of models undoubtedly dulls the imagination of the pupil and 
 causes him to lose the ability to visualize the propositions. Yet, 
 there is no objection to placing models in the hands of the pupil, 
 provided the models are simple ones, constructed of such ma- 
 terials as corks, knitting needles, sticks, or clay. If the pupil 
 of his own free will wishes to make cardboard models at home, 
 he should be encouraged ; but to require this is decidedly un- 
 profitable. Such work is a waste of time, turns mathematics 
 into handwork, and is exceedingly laborious for those pupils 
 who are not skillful with their hands. Yet there are some very 
 worthy authorities who are in favor of having the pupil perform 
 this construction during the class hour. Undoubtedly there are 
 instances where putting together a model brings new points 
 to the notice of the pupil. On the other hand, apparatus should 
 not stand in the way of getting a clear mathematical insight into 
 the problem ; just as in art the production should not be too 
 realistic, so in mathematics there must be some work left for 
 the imagination. At the end of this discussion an extensive list 
 of models is incorporated in the report.
 
 30 Teachers College Record [96 
 
 There is a general agreement upon the practical applications 
 in mathematics. Where there are only a few who warn us 
 against the overuse of these examples, there are many who urge 
 that illustrations be taken from physics, chemistry, surveying, 
 astronomy, and geography. Some claim that field work should 
 occupy two-thirds of the time allotted to mathematics. Others 
 are of the opinion that there is too large a number of students, 
 too great a scarcity of time and instruments, and too few 
 teachers for such work ; otherwise it would be most profit- 
 able to have much practice in the open air. Everywhere 
 there seems to be a demand for stronger emphasis on the appli- 
 cations. The teachers assert that the only way to lead to the 
 theoretical is by means of the practical, while the patrons are 
 calling for work of practical value. This method of teaching 
 may be used to advantage at certain stages of the pupil's progress, 
 but its general adoption is apt to frustrate the very aims and 
 purposes of the course of mathematics in the secondary schools. 
 If too much field work is introduced and too many applications 
 are used, there is danger that mathematical instruction may pass 
 entirely into the domain of the natural sciences and that the 
 vigor of sound mathematical reasoning be lost. 
 
 Agitation for the separation of the sciences into two main 
 groups, namely, a mathematical-physical group and a chemical- 
 biological group, is current. Many teachers of mathematics 
 are in favor of such a division. As our courses are arranged 
 to-day, such a classification is not natural and would not furnish 
 adequate preparation for the courses in physics and chemistry at 
 the universities. A valid claim might also be made that mathe- 
 matics is related to geography, mineralogy, or biology. Applied 
 mathematics is the natural companion of pure mathematics and 
 should be kept in that department. 
 
 The recommendations of the commission set forth in this 
 report are conservative, yet they show that the suggestions of 
 the reformers have had their influence and have served to enrich 
 and improve mathematical instruction. Subjects worthy of in- 
 terest are the early introduction of the function idea, the extent 
 to which the use of models shall be carried, and the use of the 
 text-book. We find that in Germany, as in America, a general
 
 97] The Present Teaching of Mathematics in Germany 31 
 
 demand for practical applications is voiced by all interested, 
 and that teachers of pure mathematics are anxious that this 
 demand shall not spoil the presentation of the subject. In gen- 
 eral, the report shows that mathematical instruction in Germany 
 is strong in every particular; that the demand for improvement 
 is carefully discussed by educational authorities; and that the 
 needs are very similar to the needs of mathematical instruction 
 in America.
 
 CHAPTER V 
 THE SECONDARY SCHOOLS OF WURTTEMBERG 1 
 
 Isidore Skolnick 
 
 This report is divided into seven chapters. The first chapter 
 treats of the various schools in Wiirttemberg. The second and 
 third chapters give an account of the mathematics course offered 
 and of the various reforms which have been proposed in the 
 Gymnasium and Realgymnasium respectively. The fourth re- 
 views the mathematics of the Madchenschulen and hohere Lehr- 
 erinnenseminar of Stuttgart. In chapter five the text-books on 
 arithmetic, algebra, geometry, stereometry, and other subjects 
 are reviewed. The sixth chapter gives an account of the exam- 
 inations, while the seventh and last chapter contains facts con- 
 cerning the preparation of teachers for the high schools. 
 
 TYPES OF SCHOOLS 
 
 The three types of schools, the Gymnasium, the Realgym- 
 nasium, and the Oberrealschule, are found in the kingdom of 
 Wiirttemberg and vicinity. The arrangement of classes in these 
 schools differs slightly from that of the North German schools 
 and the comparison is given in the table which follows. 
 
 WURTTEMBERG 
 
 
 NORTH GEF 
 
 fCb 
 
 188 11 
 
 
 Lower division { 
 
 11 
 
 Unterstufe 
 
 1 
 
 mj 
 
 
 
 IV] 
 
 
 Middle division ] 
 
 V > 
 
 Mittelstufe 
 
 [ 
 
 VI J 
 
 
 c 
 
 VII] 
 
 
 Higher division { 
 
 VIII \ 
 
 Oberstufe 
 
 I 
 
 IX J 
 
 
 1 Der Mathematische Unterricht an den Hoheren Schulen nach Organi- 
 sation, Lehrstoff und Lehrverfahren und die Ausbildung der Lehramts- 
 kandidaten im Konigsreich Wurttemberg, von Dr. Erwin Geek, Leipzig 
 und Berlin, 1910. 
 
 32 [ 9 8
 
 99 j The Present Teaching of Mathematics in Germany 33 
 
 Shortly before 1903, Class I was divided into two parts, thus 
 making ten classes instead of nine. The average age of the 
 students on entering the first class is eight years, which means 
 that they graduate at the age of eighteen. In certain sections of 
 Wurttemberg there are elementary schools having a two- and 
 three-year course. 
 
 In 1787 the first Realschule was founded in Niirtingen, and 
 in 1793 additional schools of this type were recognized by the 
 government. The Realschulen are attended by the older students, 
 who take such courses as will prepare them to become practical 
 business men. There are also a number of schools that give 
 theological courses. 
 
 The Biirgerschulen of Stuttgart are midway between the Real- 
 schulen and the Volkschulen (common schools). They differ 
 from the Realschulen in that English is elective, that there are 
 eight classes instead of ten, and that the lowest class may be en- 
 tered by pupils who have attained the age of six years. They 
 are primarily for those who are to enter the commercial field. 
 
 The high schools for girls have ten classes, the girls entering 
 at the age of six. At Stuttgart there is a higher seminar for 
 girls connected with their high school, and, in addition to this, 
 an excellent school called the Madchengymnasium. In other 
 localities, where there are no high schools provided for the pur- 
 pose, girls are permitted to attend the higher boys' schools, the 
 Knabenschulen. 
 
 A statement of the schools of Wurttemberg tabulated as to 
 types and number is here given : 
 
 A. Humanistic schools. 
 
 1. Gymnasien (14) and Theological Seminaries (4). 
 
 2. Progymnasien with upper classes (5). 
 
 3. Landschulen (48). 
 
 B. Realgymnasien. 
 
 1. Realgymnasien, all classes (5). 
 
 2. Real progymnasien, with two upper classes (8). 
 
 C. Realschulen. 
 
 1. Oberrealschulen, all classes (12). 
 
 2. Realschulen, with one upper class (17); with two upper classes (4). 
 
 3. Landschulen (68). 
 
 4. Burgerschulen, in Stuttgart, (2). 
 
 D. Madschenschulen, public (17); private (6). 
 
 Teachers' Seminar for Women in Stuttgart.
 
 34 Teachers College Record [100 
 
 THE COURSE IN MATHEMATICS IN A GYMNASIUM PRIOR 
 TO THE REFORMS OF 1891 
 
 Class I Addition, subtraction, multiplication, and division by a two 
 figure divisor; tables of measures, weights, and time with 
 problems involving the same. 
 
 Class II Easy decimals and a little percentage with occasionally some 
 work in proportion. 
 
 Class III Factoring, prime factors and fractions. 
 
 Class IV-V Fractions continued, including complex fractions, and discount. 
 
 Class VI Algebra: fundamental operations, simple equations of the first 
 degree, theory of indices and, proportion. Geometry: dis- 
 cussion of lines, angles and triangles, and propositions on the 
 parallelogram and circle, the teacher dictating the proofs. 
 
 Class VII Algebra: solution of simple equations and equations of the 
 first degree in more than one unknown. Geometry: plane 
 geometry continued, with algebraic interpretation. Trig- 
 onometry: fundamental notions of trigonometric functions 
 and some formulas. 
 
 Class VIII Algebra: study and use of logarithms, affected quadratic equa- 
 tions, simultaneous quadratic equations, and theory of 
 experiments. 
 
 Class IX Algebra: binomial theorem. 
 
 MATHEMATICS IN THE REALGYMNASIUM AND REALSCHULE 
 
 About the year 1810 the students of the Stuttgart Gymnasium 
 expressed dissatisfaction with the existing course of study, claim- 
 ing that the Latin and Greek to which they were obliged to 
 devote a considerable portion of their time, did not fit them for 
 their subsequent positions in life. They were to become officers 
 and " Hofleute " and therefore they needed a more extensive 
 knowledge of mathematics than of Greek. In response to this 
 demand the curriculum was divided into groups A and B, group 
 A retaining the Latin and Greek and group B having an increased 
 amount of mathematics and modern language. These changes 
 were made effective in classes VI and VII of the Obergymnasium. 
 The students displayed great ability and enthusiasm for the work 
 in modern languages and mathematics, and in 1859 there was a 
 special teacher giving instruction in algebra, geometry and 
 geometric drawing. 
 
 As early as 1865 the need for men trained in the various 
 lines of scientific work began to be felt very keenly, and the 
 outcome of it was a clear division of the schools, the Realgym-
 
 loi] The Present Teaching of Mathematics in Germany 35 
 
 nasium becoming separate and distinct from the old Gymnasium, 
 and developing into a scientific school. In 1871 the first teachers' 
 examinations were held in that school, and in the same year a 
 new program was formulated which offered to the student the 
 desired and necessary subjects in mathematics. Mathematics 
 was entirely eliminated from classes VI and VII and in classes 
 VIII and IX analytic geometry, spherical trigonometry, and 
 mathematical geography were given. In the Stuttgart Realgym- 
 nasium, the proportion of the students' time (expressed in the 
 number of hours out of the total number spent on all subjects) 
 which was devoted to mathematics in the various classes was as 
 follows: I, 4 out of 25; II, 4 out of 25; III, 4 out of 31; 
 IV, 4 out of 31 ; V, 5 out of 32; VI, 5 out of 33; VII, 7 out 
 of 33; VIII, 17 out of 34; IX, 8 out of 34. 
 
 In the Stuttgart Realschule the course extended over four 
 years. The purpose of the school was to prepare men for 
 scientific and technical work, and almost one-half of the students' 
 time was devoted to work in mathematics. This work included 
 all of the subjects from arithmetic through spherical trigonometry. 
 
 The reforms of 1904 and 1906 made the Oberrealschulen of 
 Wiirttemberg true technical schools. In the upper classes the 
 number of hours devoted to mathematics was decreased and the 
 number given to the other sciences was increased. 
 
 THE HIGHER GIRLS' SEMINAR 
 
 The purpose of the Girls' Seminar in Stuttgart is to offer 
 courses to girls who are preparing to teach in the lower and 
 middle divisions of the girls' high schools. The course extends 
 over a period of three years and is encyclopedic in nature, with 
 the emphasis put upon pedagogy. The work of the first two 
 years is given in detail later. The third year's work is mainly 
 a specific preparation for what is called the " leaving examina- 
 tion," the successful passing of which entitles the candidate to 
 teach. The subjects in which they are examined are religion, 
 German, French, English, history, geography, natural history, 
 mathematics, hygiene and pedagogy. 
 
 We find in the three years' course in mathematics of this school 
 an interesting attempt to give a general survey of the field of the
 
 36 Teachers College Record [102 
 
 less advanced secondary mathematics. It presupposes the aver- 
 age amount of preparation in elementary mathematics and in- 
 cludes work in algebra, geometry, trigonometry, and stereometry 
 with which is combined geometric drawing. Although the Eucli- 
 dean geometry is strictly followed, they enter into discussions 
 concerning the reorganization of the subject matter as to the 
 number and sequence of theorems. Some topics as, for instance, 
 inscribed and circumscribed polygons and the theory of limits, 
 which are omitted from the ordinary girls' high school course, 
 are given here. The concept of function and the graphic repre- 
 sentation of algebraic functions claim their share of the students' 
 attention. The work in trigonometry includes the solution of the 
 right triangle, and practical problems are given which afford 
 practice in the use of four-place logarithmic tables. Spherical 
 trigonometry is not emphasized. Stereometry is touched upon 
 briefly, the work being confined to a study of the prism, pyramid, 
 cylinder and sphere. The texts used are Speeker's " Geometry '*' 
 and Bardy-Hartenstein's "Algebra." For the classes in trigo- 
 nometry and stereometry there are no text-books, manuscripts 
 being used instead. A very little work in physics, supplemented 
 by just enough chemistry to make the physics clear, constitutes 
 the course in these branches of science. 
 
 At the termination of the course the students are given an 
 examination which is divided into two parts. The first part, 
 given at the end of the second year, is a test in rapidity as well 
 as in content. The second part determines the qualification of 
 the candidate for high school positions. 
 
 THE ToCHTERSCHULEN 
 
 The Tochterschulen are not strictly secondary schools but 
 there is a seminar in connection with the higher classes in which 
 secondary subjects are taught. These schools are either public 
 or private and contain ten classes. In the lower classes instruc- 
 tion is offered in what is termed biirgerlicher Rechnen, but in 
 the upper classes or the seminar, algebra and geometry are taught 
 by a high-school teacher. 
 
 In 1903 the public school board decided that the standard of 
 these girls' schools should be raised, and accordingly a regular
 
 103] The Present Teaching of Mathematics in Germany 37 
 
 and fairly extensive mathematics course was incorporated into 
 the school curriculum. The complaint which one hears to-day in 
 connection with these schools is that the teachers are of the old, 
 non-progressive type and that they are slow to adopt any of the 
 reforms in mathematics teaching. Considerable attention is paid 
 to the work in arithmetic which includes a large amount of busi- 
 ness calculation, four hours per week being devoted to the work 
 in the first five classes, three hours in the sixth and seventh, 
 and two in the last three classes. Special emphasis is put 
 upon mental work. In classes IX and X, three hours are given 
 to geometry. The more important propositions of plane geometry 
 are studied and work in geometric drawings, exercises and con- 
 structions is required. The course in algebra is not extensive. 
 
 MATHEMATICS TEXT-BOOKS IN GERMANY 
 
 " Die Methodischen Grammatik des Schulrechnens," a series 
 of text-books which cover the subject from elementary arith- 
 metic up through advanced mathematics, is used extensively 
 throughout the high schools. It is especially helpful to the 
 teacher in presenting the subject, and contains practical and con- 
 crete problems connected with the daily life of the pupil. Of the 
 texts in geometry and stereometry Spiekersche's " Lehrbuch der 
 ebenen Geometric" (1908) is a popular book. During the first 
 two years of the high school Mahler's text is used. In sterom- 
 etry the texts of Kommerell, Hauck, Tubingen, and Bragg 
 (1908) are studied. In trigonometry, Burklen's "Lehrbuch" 
 (Heilbronn, 1897) is used. Students , have the privilege of 
 selecting the text to be used for the study of spherical trigonome- 
 try. In analytics a number from the Sammlung Goschen is used. 
 Other mathematics is studied from manuscripts. 
 
 QUALIFICATION OF TEACHERS OF SECONDARY SCHOOLS 
 
 The custom of giving teachers' examinations was inaugurated 
 in 1846. Examinations were given in German, French, nature- 
 study, literature, geography, and mathematics, the last named 
 including the following topics: arithmetic, algebra, geometry, 
 stereometry, advanced trigonometry, and practical geometry, e.g., 
 the use of measuring instruments. The candidate who wished
 
 38 Teachers College Record [104 
 
 to qualify for the, Oberrealschule had to pass examinations in 
 spherical trigonometry, analytics, calculus, practical geometry 
 (advanced), mechanics, and machine construction and design. 
 He must also have had a two years' course in some polytechnic 
 school and have attended a university for one year. He was 
 required to take an oral examination in the presence of the pro- 
 fessor in whose charge he was to be placed, and to pass an 
 examination on the teaching of mathematics. The mathematics 
 teacher was required to pass examinations, not only in the sub- 
 jects which he expected to teach, but in other subjects as well, 
 as he was expected to be able to take the place of other teachers 
 in case of absence. Not more than two opportunities to take 
 the examinations were allowed the prospective teacher. 
 
 In 1907 a new system of examinations affecting the third from 
 the upper class of the Wurttemberg schools was inaugurated. 
 This examination was required of everyone and covered all sub- 
 jects then being pursued. The examinations were very rigid and 
 consisted of both oral and written work. It was made possible 
 for one who passed an excellent written examination to be ex- 
 cused from the oral one.
 
 CHAPTER VI 
 THE SECONDARY SCHOOLS OF BAVARIA 1 
 
 M. J. Leventhal 
 
 The object of this report by Professor Wieleitner is to give a 
 survey of the evolution of mathematics in the Bavarian high 
 schools, together with additional information concerning the 
 training and extension work of teachers in Bavarian Gymnasien, 
 and in the humanistic schools in particular. 
 
 The work done in the latter type of school may suggest im- 
 provement of our own program. The school year in Bavaria is 
 only a trifle longer than ours, extending from September i8th 
 to July I4th, yet the work done is far superior to ours, both in 
 character and content. 
 
 With regard to the correlation of mathematics with other allied 
 sciences, Bavaria is still undecided. Among the instructors, 
 there are only two who advocate a combination of all the natural 
 science departments. Professor Wieleitner is of the opinion that 
 the teacher should teach the same class both mathematics and 
 physics, although he confesses that few of the teachers know 
 mathematics and physics equally well. , It is generally recog- 
 nized that the physicist is apt to treat his mathematics too 
 empirically, and the mathematician to treat his too mathemat- 
 ically, but nevertheless it is the general feeling that this is 
 better than to have two separate capable instructors teach the 
 two subjects without any mutual reference to each other. 
 
 At present there are two scientific groups in the Bavarian 
 schools: (i) mathematics and physics, (2) chemistry and biology. 
 The arranging of these two groups is considered a great step 
 forward. For a long period in Bavarian Gymnasien the old- 
 
 x Der Mathematische Unterricht an den Hoheren Lehranstalten sowie 
 Ausbildung und Fortbildung der Lehrkrafte im Konigreich Bayern, von 
 Dr. Heinrich Wieleitner, Leipzig and Berlin, igro. 
 
 105] 39
 
 4O Teachers College Record [106 
 
 time master, trained only as a philologist, taught mathematics, 
 physics, French, and theology, besides his Latin and Greek, and 
 the establishing of these groups put an end to the old regime. 
 
 THE HUMANISTIC SCHOOL 
 
 This type of school deserves our special attention, for it is 
 the type which may best be compared to our average high 
 school. In the colleges, it would correspond to the so-called 
 " arts course." From the Gymnasium, jurists, theologists, phil- 
 ologists, physicians, and officers get their mathematical knowl- 
 edge, and in the early days this knowledge was very slight. 
 
 Up to 1901 all reforms and suggestions that were proposed 
 seemed merely to cause retrogression in the school program, and 
 to foster still greater conservatism in the system. In the year 
 1901, however, a definite improvement is noticeable, and conse- 
 quently we shall at first confine our attention to the advance that 
 was made in that year. We shall then examine the new sug- 
 gestions and reforms proposed by Bavarian educators in the 
 attempt to ameliorate the status of mathematics in the human- 
 istic schools. 
 
 The classes in these schools are numbered, as is the case in most 
 German high schools. For our purposes we shall use the num- 
 bers from i to 9 to indicate the successive school years as usual, 
 beginning with the lowest ( I ) . For the sake of brevity, we shall 
 not outline the program in full, but shall note only the con- 
 spicuous features. It will be our task, then, to determine 
 whether the American teachers of mathematics can gain any- 
 thing from the experience of the schools in Bavaria. 
 
 A conspicuous feature of the program of the Bavarian schools, 
 which can scarcely escape our attention, is their emphasis upon 
 mental work. The Bavarian student can work problems men- 
 tally that our American high-school boys would find hard to 
 solve even with the use of pencil and paper. This mental 
 training makes the former more acute, and makes him a swifter 
 and more accurate calculator. Besides, despite our modern psy- 
 chologists, it is claimed that it improves his power of memory. 
 The statement, " I'll look it up," so common with American stu- 
 dents is not in vogue in Bavarian humanistic schools.
 
 107] The Present Teaching of Mathematics in Germany 41 
 
 Parentheses are taken up in the fourth school year. When we 
 come to percentage and write the formula A = B (i + R), our 
 students are puzzled to know the significance of those " curved 
 lines." The Bavarian child understands them as soon as he 
 takes up the multiplication of integers. In algebra not much 
 has to be said about the subject. The pupil learns very early to 
 operate with x, y, and a, as he does with 5, 10, and 12. 
 
 In the fifth school year, common and decimal fractions are 
 taken up simultaneously. The student is taught to see clearly 
 that the decimal is nothing but a new expression for the ordinary 
 fraction. He sees the identity of an answer when given in 
 decimal form, in the operation of multiplication, for instance, 
 with the answer, when all the elements of the operation are 
 reduced to common fractional forms. 
 
 Having completed the work in percentage, alligation, and 
 interest, algebra is begun in the eighth school year. The work 
 is such as we have in the last year of some of our best ele- 
 mentary schools. At this time they also begin geometry. It 
 does not at all coincide with our mensuration, since they take up 
 congruence of triangles, loci, and quadrilaterals and their prop- 
 erties. 
 
 With the next year, we find the beginning of what corre- 
 sponds to our high-school course. In algebra the work includes 
 linear equations, such proportion as is needed for the study 
 of the similarity of triangles, equations involving two or three 
 unknowns, theory of exponents, extraction of the square root, 
 and quadratics involving one variable. In geometry we find 
 such topics as the following: similarity, relations and measure- 
 ment of rectilinear figures, equivalence, computation exercises, 
 and construction problems. 
 
 We should notice in this arrangement of the work that a 
 program bringing out a perfect correlation of algebra and geom- 
 etry is possible, but that no fusion of the two branches in one 
 is attempted. Proportion in algebra aids the student in under- 
 standing similarity in geometry, and similar figures show him the 
 practical utility of the " Rule of three." Furthermore, a few 
 of the laws of algebra, often without much meaning to the stu- 
 dent, become intelligible and full of significance when seen from
 
 42 Teachers College Record [108 
 
 the standpoint of geometry. Correlation without destruction 
 seems the dominant idea. 
 
 Logarithms, postponed in our course to the last high-school 
 year, are taken in the tenth year. Why should not the student 
 be able to use this handy tool, the logarithmic table, to work out 
 complicated computations, which are to him so laborious? The 
 Bavarian teachers continue their plane geometry in the later 
 years, with especial emphasis upon algebraic-geometric exercises, 
 and with a thorough study of the mensuration of plane surfaces. 
 As soon as the use of the logarithmic tables is understood, the 
 student's mind is deemed mature enough for the study of 
 trigonometry. Important goniometric formulas are then studied, 
 and the measurement of solid angles and crystals is particularly 
 emphasized. 
 
 Thus the numerical work prepares the student for computing 
 the lateral areas and volumes of the ordinary solids in the eleventh 
 school year. In general, the student is advanced far enough to 
 be capable of the appreciation of the more difficult theorems in 
 solid geometry, such as Euler's theorem, and the prismatoid 
 formula. 
 
 It is thought that the student has quite enough of analytic and 
 synthetic work to begin his analytic geometry in the secondary 
 school. At present this subject is treated in various schools 
 according to a set program. The course is intended merely to 
 give the pupil a training in the fundamental notions and methods 
 of analytics, being limited to a study of the co-ordinate system 
 and its applications in physics and statistics. 
 
 It would be interesting to mention in some detail certain of the 
 topics treated in the course, but we have space for only a single 
 illustration. For example, the following equations are consid- 
 ered, their meaning explained, and their graphs drawn and fully 
 discussed : 
 
 y' = r', (x a)' + (y b)=r, 
 x=rcosa, y =rsina, 
 
 x l y* 
 
 and y = sin x.
 
 109] The Present Teaching of Mathematics in Germany 43 
 
 In the twelfth school year, it is the aim to collect all the theory, 
 show the connection of formulas, and demonstrate their use in 
 the physical sciences. On this account mathematical geography 
 is taken up, together with Kepler's laws, Newton's law of gravi- 
 tation, and the use of co-ordinates in the determination of the 
 position of stars. A good deal of mathematical physics is intro- 
 duced in this connection. 
 
 Under the new plan proposed, some of the calculus is to be 
 introduced in the eleventh school year, and in the following year 
 maxima and minima and a study of differential quotients are to 
 be studied. 
 
 It is also thought that it would not be too radical to introduce 
 trigonometry in the ninth or tenth year, directly after the study 
 of similarity of figures. Many relations in geometry which we 
 express to-day by proportion could still better be expressed by 
 the use of the sine and the cosine. 
 
 SOLID GEOMETRY 
 
 It has been proposed to put some of the simpler work in vol- 
 umes, points, lines, and planes a little earlier. This, however, 
 depends on the question whether it would not be more profitable 
 to treat the plane and solid geometries together. In Italy, since 
 1900, they have had the fusion system in some schools. In 
 Bavaria, however, they have no book, as yet, in which this 
 plan is worked out. In the Italian schools they have Lazzeri 
 and Bassani's " Primi Elementi di Geometria " and one or two 
 other similar works. 
 
 These reforms have been brought about by the earnest activity 
 of the various organizations of teachers. All plans that are pro- 
 posed are investigated by the Bavarian section of the Organiza- 
 tion of Teachers for the Promotion of Mathematical and Physical 
 Sciences. Much of the advance is also due to an active society 
 in Munich, the Munich Society of Teachers. Since 1907, a 
 number of similar associations have been formed, which, besides 
 investigating various reforms, have arranged for numerous 
 scientific reports and lectures. The spirit of the Bavarian teach- 
 ers is admirable, and their achievements are worthy of emulation 
 in our country.
 
 CHAPTER VII 
 THE HIGHER SCHOOLS FOR BOYS IN PRUSSIA 1 
 
 Robert King Atwell 
 
 The well-known American authority, Professor J. W. A. 
 Young, who is quoted by Dr. Lietzmann in this report, explains 
 the superiority of the German schools over the American schools 
 in this manner : " The causes of the excellence of the Prussian 
 work in mathematics may be classed under three heads : ( i ) The 
 central legislation and supervision. (2) The preparation and 
 status of the teachers. (3) The method of instruction." ' Of 
 these three contributing causes, the first may well deserve a 
 little attention at this time, since it is not discussed in any other 
 chapter. 
 
 EDUCATIONAL ADMINISTRATION 
 
 The educational work of the state is under the direction of the 
 Ministry of Spiritual, Educational and Medicinal Affairs, estab- 
 lished in 1817. Besides the Minister, there are an under-State 
 Secretary and Division Directors. In addition to these officials, 
 each division includes reporting boards and assistants selected 
 from the body of professional men and administrators. The Min- 
 ister reaches his decisions independently, since " every ministerial 
 decision is considered as coming from the Minister himself." He 
 is answerable to the King and the Diet. Since Prussia has no 
 educational statutes, the policy of higher public instruction is 
 determined by ministerial decrees. 
 
 The individual schools are not directly answerable to the 
 ministry. There are intermediate authorities, namely, the Pro- 
 vincial School Boards. There are twelve of these bodies in the 
 capital of the Prussian provinces. The Presidents are the Gen- 
 eral Presidents of the province. The Directors (or vice-presi- 
 
 1 Die Organisation des Mathematischen Unterrichts an den Hoheren 
 
 Knabenschulen in Preussen. von W. Lietzmann. Leipzig und Berlin, 1910. 
 
 * J. W. A. Young : The Teaching of Mathematics in Prussia. N. Y. 1900. 
 
 44 [no
 
 in] The Present Teaching of Mathematics in Germany 45 
 
 dents) are in some cases schoolmen, but usually administrative 
 officers. The number of members of the Provincial School 
 Boards varies from one to five, if we include only the " Dezer- 
 nenten " l 'for the higher public instruction ; to this number are 
 sometimes added pedagogical experts. 
 
 The Provincial School Boards become acquainted with the 
 schools of their districts through short visits as well as by con- 
 ducting the final examinations of the " higher secondary " insti- 
 tutions, and by granting the certificate of the Lower Secondary 
 School which indicates that the candidate's final examination 
 has been passed, and entitles him to a diminution of one year 
 of military service. To each member of the Board a certain 
 number of students of his district is assigned, to be under his 
 special charge. He watches these students during their entire 
 course. These lists are revised every three or four years. 
 
 All important matters, such as the maintenance of instruction, 
 the selection of the subject matter of the readings in the lan- 
 guages, the introduction of text-books, and the like, are subject 
 to the ratification of the Provincial School Boards. They also 
 appoint the teachers in the royal institutions and confirm them 
 in the city institutions. 
 
 There are numerous non-state schools, mainly city high 
 schools. The private or the religious schools of Prussia are 
 not considered in this report. In the west, these schools for the 
 most part have " guardians " or trustees ; in the east, the magis- 
 trate exercises his rule directly; in the large cities, a technical 
 assistant is assigned to the city high schools. The guardians and 
 magistrates exercise the power of selecting the directors and 
 teachers, as well as of preparing the budget. The budget is 
 afterward referred to the board appointed by the city. 
 
 The administration of the various schools is in the hands of 
 the directors who are not only administrative officers but teach- 
 ers as well, required to teach a certain number of hours. 
 
 Concerning the relation of the director to his teachers no 
 universal rules have been laid down, though some such regula- 
 tions are now being considered. In fact, in all the provinces 
 there is some " service instruction," by which it is very often 
 
 1 Appointees.
 
 46 Teachers College Record [112 
 
 stated that the director is omnipotent and the teacher has no 
 rights; but these strict decrees frequently exist only on paper. 
 The relations between the director and the teaching staff are 
 largely determined by the personality of those concerned, and 
 not by official decrees. 
 
 Every four years, in all the provinces of Prussia except 
 Brandenburg (as well as Hessen-Nassau until very recently), the 
 directors of the higher boya' schools meet in the General Assem- 
 bly of Directors, which is attended also by the members of the 
 provincial school board and councillors from the ministry. 
 
 THE REFORM MOVEMENT 
 
 As early as 1899 tne seventh Directors' Assembly of the 
 Rhine sought expert advice upon the question: What sugges- 
 tions for the bettering of mathematical instruction, recently pub- 
 lished, deserve to be put in practice in the higher schools ? 
 
 The reform movement in mathematical instruction finds its 
 most definite expression in the so-called Meran and Stuttgart 
 " Proposals " of the Commission on Instruction, appointed by 
 the Association of German Natural Philosophers and Physicians. 
 The first conference of this body was held in 1900. After that 
 it met yearly, but the first meeting considered worthy of any 
 extended report was held in 1908. 
 
 The year 1908 is noteworthy for the appointment of the Ger- 
 man Commission for Mathematics and Science Instruction. The 
 commission included among its members such prominent mathe- 
 maticians as Professors Gutzmer, Klein, Poske, Schotten, 
 Stackel, Thaer, and Treutlein. 
 
 Of the questions discussed by this commission the following 
 are especially worthy of note: 
 
 1. A report concerning the mathematical instruction in the 
 " higher institutions of more classes." Meran, 1905. 
 
 2. The mathematical and natural science instruction in Re- 
 formschulen. Stuttgart, 1906. 
 
 3. The mathematical and natural science instruction in the six- 
 class Realstalten. Stuttgart, 1906. 
 
 The suggestions of the Commission on Instruction embody two 
 phases of the mathematical instruction in the higher schools :
 
 113] The Present Teaching of Mathematics in Germany 47 
 
 1. The strengthening of the power of spatial conception. 
 
 2. Training in the habit of thinking in functions. 
 
 In their various ways the members of the Commission seek 
 to emphasize the following ideals : 
 
 1. To adapt the course more than formerly to the natural 
 steps of mental development. This psychological principle mani- 
 fests itself in the emphasis upon propaedeutic instruction in 
 arithmetic and geometry, in the demand for a gradual transition 
 from intuitive to deductive treatment. 
 
 2. " To bring to the attention of teachers of mathematics the 
 possible solution of the problems of the visible world around 
 us." This utilitarian principle shows its influence in the search 
 for applications of mathematics. 
 
 3. To make the subject matter of the course within itself, 
 from class to class, more and more coherent. This didactic 
 principle leads to the concentration of the instruction in the ag- 
 gregate around one fundamental idea, the function, in the sub- 
 jects of algebra and geometry. 
 
 For the purpose of showing the influence of the reform, the 
 various secondary institutions may be divided into groups, the 
 classification being made according to their attitude toward the 
 adoption of the suggestion. For example, the classification of 
 the upper classes (oberstufe) of these institutions is as follows: 
 
 I. Those schools which refused to introduce the idea of func- 
 tion. 
 
 II. Those which believed in a late (uncertain) introduction of 
 the function concept. 
 
 a. Without calculus. 
 
 b. With differential calculus. 
 
 c. With differential and integral calculus. 
 
 III. Those which preferred a gradual (regular) introduction of 
 the function concept. 
 
 a. Without calculus. 
 
 b. Wtih differential calculus in the Prima. 
 
 c. With differential and integral calculus in the Prima. 
 
 d. With integral calculus already in the upper half 
 of the class below the Prima.
 
 48 Teachers College Record [114 
 
 In making up this list, the programs of over six hundred insti- 
 tutions were examined. There were but few institutions which 
 were radically " reformed," while the remaining institutions were 
 about equally divided between those that adopted the reform 
 to a moderate extent, and those that ignored the reform alto- 
 gether. 
 
 The question as to how far this most recent reform agitation 
 has penetrated into the schools can be estimated roughly from 
 another grouping of institutions which includes only those which 
 show changes in the programs. 
 
 On the basis of the Easter program of 1909, a list has been 
 arranged which gives merely an approximation, since only the 
 programs dispatched in August, 1909, could be considered. Ac- 
 cording to these data, the introduction of the idea of function 
 was noted in twenty-four Gymnasien, thirty-five Realgymnasien, 
 thirty-four Oberrealschulen, and five Realschulen, making a total 
 of ninety-eight institutions. 
 
 The recent additions to the requirements in mathematics are 
 nothing new in themselves. During the last twenty or thirty 
 years, certain far-sighted men have already done what the Meran 
 Proposal calls for. Proof of this will be found in numerous 
 instances by a later historian of the Reform Movement, who, 
 being removed by a greater interval of time from these move- 
 ments, which are at present in such a state of transition, will be 
 able to make a completely objective judgment. 
 
 The psychological principle referred to above has had zealous 
 advocates before the present agitation for reform, to name only 
 Treutlein in Baden and Hofler in Austria. There are many 
 mathematicians, however, who do not agree on this point, al- 
 though otherwise heartily in sympathy with the reform move- 
 ment. 
 
 The suggestions embodied in these " Proposals " have been 
 made before. Among the instances which might be cited is that 
 apparently almost forgotten report of A. von Oettingen concern- 
 ing the mathematical instruction in the schools of Dorpat. In it 
 we find the following requirements: 1 (i) The introduction of 
 
 1 Report on the anniversary of the founding of the University of 
 Dorpat, 1873.
 
 115] The Present Teaching of Mathematics in Germany 49 
 
 the relations of variable quantities ; in short, the idea of function. 
 (2) The bare elements of analytic geometry as far as the calculus. 
 However, in the opinion of Dr. Lietzmann, it is entirely wrong 
 to assert that the proposed reforms in the instruction in mathe- 
 matics teach nothing new, that what is now desired was long 
 ago brought out by others. Hofler once pointed out to some of 
 the men of greatest influence in mathematical circles that the 
 highest compliment that can be paid to the reform movement 
 is that it contains no items which are fundamentally new, and 
 that no matter how many such forerunners may be found, it 
 is the harmonizing of all these proposals, which formerly were 
 often sharply opposed, into one powerful impulse, as well as 
 the co-operation of so many forceful personalities, which makes 
 this movement one for which no analogy can be found in the 
 history of mathematics.
 
 CHAPTER VIII 
 
 THE SECONDARY SCHOOLS OF ELSASS AND 
 LOTHRINGEN 1 
 
 Maurice Levine 
 
 In the first chapter Dr. Wirz describes the present organiza- 
 tion of the higher schools in Elsass-Lothringen (Alsace-Lor- 
 raine). This is followed by a historical and critical survey of 
 the development of instruction in mathematics, especially with 
 regard to the curriculum, from the time of the French control 
 in 1870 to the present day. The third chapter deals with the 
 methods of instruction employed at the present time. The part 
 that text-books play, the question of propaedeutic instruction, the 
 use of models, and practical exercises, are all discussed. The 
 fourth chapter is devoted to a discussion of the reform move- 
 ment in Elsass-Lothringen. The author gives the opinions of 
 various colleagues as to the use and scope of the function-con- 
 cept, the introduction of the differential and integral calculus in 
 the secondary school, the cutting down of the formal operations, 
 the simultaneous treatment of planimetry and stereometry, 
 geometrical drawing, the use of historical material, the instruc- 
 tion in the upper classes, and the final examinations. The last 
 chapter discusses the preparation of teachers of mathematics in 
 the higher schools. 
 
 ORGANIZATION 
 
 The higher schools of Elsass-Lothringen are under the super- 
 vision of a central board, the head of which holds a separate 
 seat in the ministry. With him are associated three assistants 
 (Oberschulrate). The board appoints the " Direktor " or presi- 
 dent of each school, a special commission of the board appoints 
 
 'Der Mathematische Unterricht an den Hoheren Knabenschulen sowie 
 die Ausbildung der Lehramtskandidaten in Elsass-Lothringen, von Pro- 
 fessor J. Wirz, Direktor der Oberrealschule in Colmar, Leipzig, 1911. 
 
 50
 
 1 1/] The Present Teaching of Mathematics in Germany 51 
 
 the instructors, and another committee has charge of the " Reife- 
 priifung." The schools of gymnasial character are organized 
 according to a definite plan. The text-books are all prescribed, 
 and the courses are limited in a general way both as to content 
 and method. 
 
 There are twenty-eight state schools (14 Gymnasien, i Pro- 
 gymnasium, 6 Oberrealschulen, 7 Realschulen) and 9 private 
 schools (6 of which are under religious control). In Novem- 
 ber, 1910, the registry of these schools numbered 10,700 stu- 
 dents (including 70 girls) who were distributed as follows: 
 5,186 at the Gymnasien, 301 at the Gymnasien with "Real" 
 courses, and 5,213 at the Realschulen and Oberrealschulen. 
 
 THE SCHOOL SYSTEM UNDER FRENCH CONTROL 
 
 There were no separate schools for humanistic and realistic 
 courses prior to the loss of the country by France (1870), but 
 the schools had a twofold object. The humanistic course was 
 nine years in length, and the classes were called huitieme, sep- 
 ticmc, sixicme, cinquieme, quatrieme, troisieme, seconde rhetorique, 
 and philosophic. This course prepared the student for the bac- 
 calaureat es-lettres. After the classe seconde, the course was 
 divided into two parts : the language-history group with classes 
 rhetorique and philosophic, and the mathematics-science group 
 with the two classes de mathematiques elementaires, to which was 
 added in a few schools the classe de mathematiques speciales. The 
 second group prepared the student for entrance to the scientific, 
 naval and polytechnic schools. 
 
 The realistic course, preparing for the practical vocations, was 
 a three years' course and started in the quatrieme. The curricula 
 varied in the different towns according to the industries that 
 were locally the most important. The courses for the two 
 groups of schools were as follows: 
 
 T. HUMANISTIC GROUP. Until the quatrieme: ordinary arith- 
 metic. 
 
 Quatrieme: some simple geometry. 
 
 Troisieme (two periods, i.e., of two consecutive hours each) : 
 factoring, prime numbers, partnership, G.C.D., fractions, 
 decimal fractions, proportion, discount, business arithmetic, lit-
 
 52 Teachers College Record [118 
 
 eral counting (introduction). In geometry: theorems on tri- 
 angles, quadrilaterals, circles, similar triangles, proportion, sur- 
 face, fundamental constructions. 
 
 Scconde (two periods 1 ) : algebraic operations, equations, rela- 
 tions between algebraic and planimetric exercises. In geometry : 
 regular polygons, circles, plane figures, limits, areas of similar 
 figures, field measurements. In stereometry: plane and line, 
 polyhedrons, prisms, pyramids, areas of parallel surfaces, 
 sections. 
 
 Rhctorique (one period): In geometry: cylinders, right cone, 
 sphere, mathematical geography, maps (various projection sys- 
 tems). 
 
 Philosophic (three periods, 1st semester, two periods, 2nd 
 semester). No definite plan was prescribed. The work in this 
 class was left to the instructor, but it was usual to review the 
 work of the former classes, and to give new work in logarithms, 
 with the use of tables, and also to take up the study of similar 
 figures in space. 
 
 The work in the clause troisieme was altogether too extensive, 
 and was really a paper course, the class not being able to ac- 
 complish even half of it satisfactorily. There was very little 
 algebra throughout, and no trigonometry at all. 
 
 Classes de mathematiques elcmen-taires (five periods) : Arith- 
 metic (a term covering algebraic work with numbers) : review 
 and further development ; imaginary quantities, maxima and 
 minima for quadratics, progressions, theory of logarithms. 
 Geometry : review ; inscribed figures, original problems. Stere- 
 ometry (practically the same as in the Prussian Gymnasium) : 
 spherical triangles, ellipse (fundamental properties), definition 
 of tangent to a curve, tangent to conies, normals. Plane trigo- 
 nometry : trigonometric lines and functions, tables, triangles, dis- 
 tances and angles of inaccessible points. Descriptive geometry 
 and mechanics: only the elements of the subjects. 
 
 Classe de mathematiques spcciales (six periods) : Arithmetic 
 and algebra: review and further development; higher equations, 
 irrational numbers, incommensurability, series, combinations, 
 binomial theorem, logarithms, exponential equations, operations 
 on circular functions, development of log (i + x) and arc tan x 
 
 1 A period or classe means two consecutive hours of work.
 
 II9J The Present Teaching of Mathematics in Germany 53 
 
 
 
 in series, theory of equations, interpolation formulas. Geometry : 
 review and further development of planimetry and stereometry ; 
 conic sections, spherical triangles, polar triangles, congruence by 
 symmetry. Trigonometry: review and further development; 
 trigonometric solutions of equations of the second and third 
 degrees, spherical trigonometry, applications to geodesy, prac- 
 tical surveying. Plane analytic geometry, with the elements of 
 solid geometry. Descriptive geometry. 
 
 2. REALISTIC GROUP : First year : arithmetic, plane geometry, 
 work outdoors, line drawing, commercial arithmetic, introduction 
 to bookkeeping. Second year : commercial arithmetic, bookkeep- 
 ing, beginning algebra and solid geometry, field work. Third 
 year: completion of elementary algebra, bookkeeping, finance, 
 accounting, trigonometry, the ordinary curves, descriptive 
 geometry. 
 
 Both the mathematics-science and realistic programs were very 
 thorough on paper, but the schools were all in a confused state 
 when Germany assumed control in 1871. 
 
 GERMAN CONTROL FROM 1871 TO THE PRESENT TIME 
 The control of the schools was immediately taken out of the 
 hands of the academies and inspectors, and put in charge of 
 the governor-general. Higher school instruction was defined, 
 but it took about a year for the program to be understood. Dr. 
 Baummeister, former Gymnasium director at Strassburg, was 
 summoned to build up the system. He completely revolutionized 
 it, imported German teachers as directors and instructors, and 
 made the following changes in spite of the opposition and preju- 
 dices of the people, including the older teachers and the pupils. 
 
 (1) Introduction (from Quinta down) of the natural sciences, 
 modern languages, physics, and chemistry. The old mathematics 
 program was allowed to remain unchanged for a time. 
 
 (2) Creation of the Landesschulkonferenz. 
 
 (3) Establishment of a definite program. 
 
 (a) Latin became obligatory throughout the course of the 
 Gymnasium. 
 
 (b) The humanistic and realistic groups were made Prussian 
 instead of French. The two groups divided in the Quarta. The 
 humanistic course, the Gymnasium, put emphasis on Latin and
 
 54 
 
 Teachers College Record 
 
 [120 
 
 Greek, while the realistic course, the Realgymnasium, empha- 
 sized mathematics and the sciences. 
 
 VI 
 
 r. 
 
 V 
 
 Gymnasium Latin and Greek 
 
 Lat. and French. 
 
 Realgymnasium Latin decreased, mathematics, English and 
 natural science increased. 
 
 In 1872, the Abiturientenexamen were introduced and were 
 almost exactly like those in Prussia. The reform in this year 
 required the students in the Gymnasium to hand in problems 
 in arithmetic and algebra fortnightly, while those in the Real- 
 gymnasium had to hand them in weekly. 
 
 Between 1873 and 1878, many important reforms were intro- 
 duced. All primary and secondary schools were placed under 
 the absolute control of the state. The three schools, Gymnasium 
 (9 years), Realgymnasium (9 years), and Realschule (7 years) 
 were defined, but all were for the time being classed as Gym- 
 nasien. The number of hours per week allowed to mathematics 
 were as follows: 
 
 CLASS No. OF'Houas 
 Gymnasium All 3-4 
 
 f VI-V 3-i 
 
 Realgymnasium \ VI-III 6 
 
 II-I 5 
 
 VI-V 4 
 
 Realachule 
 
 IV-I 
 
 
 
 The examination in mathematics called for the solutions of prob- 
 lems in all the different fields studied. 
 
 In 1882, Governor Manteuffel called a commission of educa- 
 tors and medical experts to consider whether or not too much 
 work was being demanded of the pupils. As a result of this 
 commission's work, the following changes were made : 
 
 ( i ) The higher schools were grouped as follows : 
 
 (a) Gymnasium (9 years), Progymnasium (6 years) 
 
 and Lateinschule (3 years). 
 
 (b) Realschule.
 
 i2i ] The Present Teaching of Mathematics in Germany 55 
 
 (2) The number of hours of instruction was cut down to the 
 
 following : 
 
 In VI and V, 27-28 hours weekly. 
 
 In IV and III, 30 hours weekly. 
 
 In the other classes, 32-34 hours weekly. 
 
 (3) The number of hours of home work was limited: 
 
 For VI and V, 8 hours weekly. 
 For IV and III, 12 hours weekly. 
 For II and I, 12-18 hours weekly. 
 
 (4) The program was shortened considerably. In mathe- 
 matics, the Prima students could elect spherical trigonometry and 
 analytic and descriptive geometry. 
 
 Until 1898, the number of hours in mathematics was grad- 
 ually increased with each new reform. 
 
 In 1898, the mathematics to be taught in each class of the 
 Gymnasium, Realgymnasium, and Realschule was sharply de- 
 fined, and the program is now in use, except for a few changes. 
 It is given below in full. 
 
 PROGRAM OF THE GYMNASIUM AND REALGYMNASIUM 
 
 SEXTA. Four fundamental operations with known and un- 
 known integers ; introduction to the German tables (the metric 
 system) ; time reckoning; reduction. 
 
 QUINTA. Common fractions ; short division ; long division ; 
 operations with common fractions ; reduction of fractions. 
 
 QUARTA. Arithmetic: decimal fractions, percentage, and prac- 
 tical examples. Geometry: the elements of planimetry ending 
 with the fourth congruence theorem ; elementary exercises. 
 
 UNTERTERTIA. Algebra : the four fundamental operations with 
 algebraic magnitudes ; simple equations in one unknown ; exer- 
 cises. Geometry : the parallelogram, circle, construction exercises. 
 
 OBERTERTIA. Algebra : proportion ; powers with positive inte- 
 gers ; simple equation in one and two unknowns. Geometry : 
 surfaces ; transformations ; areas of right-angled figures ; pro- 
 portionality of lines ; construction exercises. 
 
 UNTERSECUNDA. Algebra : involution and evolution ; square 
 root ; simple equations in several unknowns ; quadratic equations 
 in one unknown. Geometry: similarity: measurement of the
 
 56 Teachers College Record [122 
 
 circle; construction exercises, also exercises from the field of 
 algebraic geometry. 
 
 OBERSECUNDA. Algebra: quadratic equations with one un- 
 known ; easy equations with two unknowns ; logarithms ; arith- 
 metic and geometric progressions with applications to interest and 
 annuities. Geometry: trigonometry, right-angled triangle, sine 
 and cosine theorems ; harmonic points and rays ; similarity. 
 
 UNTERPRIMA. Algebra: difficult exercises in simultaneous 
 quadratics ; simple exercises in maxima and minima. Geometry : 
 goniometry, application to solution of triangle; stereometry; 
 planimetric exercises. 
 
 OBERPRIMA. Algebra: combinations; probability and chance; 
 binomial theorem. Geometry : mathematical geography including 
 necessary theorems of spherical trigonometry ; fundamental prop- 
 erties of the ellipse, parabola and hyperbola; exercises. 
 
 PROGRAM OF THE REALSCHULE AND OBERREALSCHULE 
 
 6. REALKLASSE. Same as Sexta in Gymnasium. 
 
 5. REALKLASSE. Same as Quinta in Gymnasium, and also the 
 fundamental operations with decimal fractions. 
 
 4. REALKLASSE. Arithmetic (4 hours in the ist semester, 3 
 hours in the 2nd semester) : decimals, changing decimals to com- 
 mon fractions ; percentage, interest, partnership and mixtures. 
 Geometry (2 hours in the ist semester, 3 hours in the 2nd sem- 
 ester) same as Quarta in the Gymnasium. 
 
 3. REALKLASSE. Arithmetic (i hour) : commercial arithmetic, 
 weights and measures. Algebra (2 hours) : four fundamental 
 operations with algebraic magnitudes ; simple equations in one 
 unknown. Geometry (2 hours) : parallelogram; circle; construc- 
 tions. 
 
 2. REALKLASSE. Algebra (3 hours) : proportion; powers and 
 real roots; square root; simple simultaneous equations in two 
 unknowns. Geometry (2 hours) : surfaces, transformations, and 
 computations; proportion (introduction to similar figures) ; con- 
 structions. 
 
 i. REALKLASSE. Algebra (2 hours) : imaginary roots, loga- 
 rithms ; quadratics in one unknown ; exponential equations. 
 Geometry (3 hours): planimetry, similarity, circle; construe-
 
 123] The Present Teaching of Mathematics in Germany 57 
 
 tions ; algebraic geometry. Trigonometry : right-angled triangle, 
 sine and cosine theorems. Stereometry : simple solids. 
 
 3. OBERREALKLASSE. Algebra : quadratic equations in two un- 
 knowns ; arithmetic and geometric progressions with applications 
 to compound interest and annuities. Geometry : trigonometry, 
 goniometry ; simple goniometric equations ; solutions of triangles ; 
 geodetic exercises ; planimetry, harmonic points and rays ; simi- 
 larity; stereometry. 
 
 2. OBERREALKLASSE. Algebra : exercises in maxima and min- 
 ima; arithmetic series of higher order; graphic numbers; com- 
 binations ; probability and error ; higher equations in the quad- 
 ratic form; cubics. Geometry: spherical trigonometry with ap- 
 plications to mathematical geography and geodesy; analytic 
 geometry of the point, straight line, and circle. 
 
 i. OBERREALKLASSE. Algebra: the binomial theorem, De 
 Moivre's theorem; infinite series with applications to geodesy; 
 numerical equations of higher degree. Geometry : analytic geom- 
 etry of the parabola, ellipse, hyperbola; projective geometry of 
 the conic sections. 
 
 This Lehrplan of 1898 was similar to the Prussian one. The 
 Gymnasium course included analytics and spherical trigonometry 
 with an elective in descriptive geometry, graphical statics, and 
 geodesy. Projective and descriptive geometry were required in 
 the Oberrealschule. Although the new changes were in line with 
 the reform movement, there was no calculus at all. 
 
 The last great changes were made in 1905. The higher schools 
 were divided into two groups : 
 
 I. Gymnasium (9 years), Progymnasium (6 years). 
 
 II. Realgymnasium (9 years), Oberrealschule (9 years), Real- 
 schule (6 years). 
 
 The mathematics for both the Gymnasium and Realgym- 
 nasium was the same (35 hours). The Oberrealschule did 
 much more in this field (45 hours). A few changes in the 
 curriculum may be mentioned. Plane and spherical trigonom- 
 etry, and plane analytic geometry were made complete, and choice 
 was given in the Realgymnasium between descriptive geometry 
 and freehand drawing. The differential and integral calculus is 
 still lacking, but projective geometry takes its place. The reform
 
 58 Teachers College Record [124 
 
 movement makes itself felt in the study of particular parts of 
 the calculus, such as maxima and minima, and series. 
 
 The Reifepriifung was made very much like the Prussian, 
 except that no one might be completely exempted from the oral 
 examination. The written examination in the three schools ex- 
 tended over five or six hours, and consisted of: 
 
 1. A German composition on a theme within the mental power 
 of each student; 
 
 2. Four problems from the different parts of mathematics. 
 To this was added: 
 
 (a) For the Gymnasium: translation from German to Latin 
 and from Greek to German. 
 
 (b) For the Realgymnasium : a French composition or a trans- 
 lation from Latin to German. 
 
 (c) For the Oberrealschule : a French composition and a trans- 
 lation from German to English. 
 
 It is the aim in teaching mathematics in the Gymnasium and 
 Realgymnasium to enable the student to know his algebra up to 
 the binomial theorem and quadratics, his plane geometry, plane 
 trigonometry, and solid geometry, and to apply this knowledge 
 to the solution of simple problems. In the Oberrealschule, the 
 requirements are the development of the most important series, 
 solution of the cubic equation, plane and solid geometry, plane 
 and spherical trigonometry, analytic and projective geometry of 
 the plane, and applications to solutions of problems. In this 
 respect, the Gymnasium course is considerably above our ordin- 
 ary high school course in mathematics, although in a good Amer- 
 ican high school the student may elect a little more advanced 
 algebra. The Oberrealschule goes as far in mathematics as 
 the freshman class in a good engineering school in this country, 
 except that in some schools the freshman completes his differ- 
 ential and integral calculus at the end of the first year. 
 
 METHODS OF INSTRUCTION 
 
 There are no definite text-books used, but new books cannot 
 be placed on the official ligf without the approval of the Royal 
 Minister. In geometry, the text-book is used for extension and 
 review, and for a survey of the whole field. The instructor does
 
 125] The Present Teaching of Mathematics in Germany 59 
 
 not follow the book, and in many schools there is not even a 
 nominal text-book, so that the personality of the instructor counts 
 for everything. However, many exercise books are used. 
 
 The question of a course preliminary to geometric instruction 
 is not defined in the official curriculum. The consensus of 
 opinion among the instructors is as follows. It should begin in 
 the Quinta and continue through the Quarta as a purely pro- 
 paedeutic course. The course should start from the contempla- 
 tion of the solids, seek out the fundamental properties, and de- 
 velop these and give them definite shape. There should be funda- 
 mental exercises in the use of the compasses, straightedge, and 
 protractor. Although a minority advocates a formal geometry 
 from the start, with definitions and demonstrations, the majority 
 opposes formal proofs before the Untertertia. 
 
 Models are in general use, especially in the upper classes, where 
 they find frequent application in stereometry and descriptive 
 geometry. They are usually made by the class, or by the more 
 capable pupils. The author, Dr. Wirz, wants the " inner sight " 
 developed and hence he does not think models at all necessary. 
 In fact, he feels that models do more harm than good, because 
 pupils frequently become satisfied with empirical proofs. 
 
 In regard to original exercises, most teachers oppose giving 
 them before the Untertertia, for they take too much of the pupils' 
 time without help, and the instructors have very little time to 
 make helpful corrections for a large class. Some think that 
 practical exercises should be the rule in mathematics, while others 
 wish the aim to be entirely logical. The author points out that 
 historically the greatest advances were made when mathematics 
 was treated independently. 
 
 The individual opinions regarding the function-concept vary 
 greatly, but the majority hold a compromise position. The 
 younger teachers advocate it most strongly, but it remains a 
 fact that the function-concept has not made much headway in 
 Elsass-Lothringen. As regards the calculus, most teachers favor 
 it for the Oberrealschule only, but desire a knowledge of the 
 symbolism and fundamental methods for the Gymnasium also. 
 
 There is a great demand for lessening the routine work in 
 the formal operations, so as to devote more time to geometry.
 
 60 Teachers College Record [126 
 
 This is opposed, and justly, by those who assert that the analytic 
 power in mathematics is just as important as the synthetic power, 
 if not more so. The cry for combining planimetry and stereom- 
 etry is opposed by a great majority on both, psychological and 
 logical grounds. Descriptive geometry is being generally re- 
 quired. 
 
 In the last decade great interest has been shown in the history 
 of mathematics, and its introduction in the last year of the Gym- 
 nasium is being advocated by all mathematicians. The question 
 as to what mathematics should be taught in the upper classes 
 is a matter of considerable discussion. There is the danger of 
 forcing too much work upon those who have no liking or apti- 
 tude for it, and, on the other hand, of giving a course too 
 mediocre for the talented student. The best suggestion seems 
 to be the division of upper classes into two groups, the first that 
 of mathematics and science, and the second that of history and 
 language. 
 
 For the Reifeprufung, described above, the following changes 
 are advocated : (a) The same problems in all the schools ; (b) 
 the choice of one large problem instead of four; (c) no oral 
 examination in some of the subjects ; (d) written examinations 
 to be made easier in certain subjects. 
 
 Most of the problems which confront Elsass-Lothringen are 
 the same as our problems, and it behooves us to consider what 
 changes should be made with the same deliberation that is shown 
 in this part of Germany. One thing that seems to be invariable 
 in all the German states is the fact that some form of geometry 
 is taught before any very serious work is undertaken in algebra. 
 This, however, is not to be misunderstood, for "Arithmetik " 
 includes our elementary algebra. 
 
 THE REQUIREMENTS FOR AND TRAINING OF TEACHERS IN 
 MATHEMATICS 
 
 The requirements for the license to teach mathematics are : 
 I. Pure mathematics. 
 
 (a) For first degree (license to teach classes up to the Unter- 
 secunda) : sound knowledge of elementary mathematics, knowl-
 
 I2/] The Present Teaching of Mathematics in Germany 61 
 
 edge of plane analytic geometry, conic sections, and the funda- 
 mental principles of the integral and differential calculus. 
 
 (b) For second degree (all classes) : in addition to above, a 
 familiarity with higher geometry, arithmetic, algebra, higher 
 analysis, and analytic mechanics, including the ability to solve 
 problems that are not too difficult. 
 
 II. Applied mathematics. A knowledge of descriptive geom- 
 etry up to and including central projection, familiarity with 
 mathematical methods connected with technical mechanics, with 
 graphical statics, and with geodesy or astronomy. 
 
 In addition to this examination in mathematics, the license for 
 first degree requires the taking of an examination in one 
 other subject, and the license for second degree in two. 
 The general examination taken by all (philosophy, pedagogy, 
 German, Latin, and religion) is held before the Wissenschaftliche 
 Kommission in Strassburg, where most of the candidates go for 
 their higher education. 
 
 The practical pedagogical training of the candidates has been 
 the same since 1871, and was taken from the Prussian system. 
 This consists in passing the Probejahr (trial year). 
 
 1 i ) 6-8 hours teaching every week. 
 
 (2) Attendance for observation in classes of the major 
 subject. 
 
 (3) Conferences and seminar (the Probe-seminar). 
 
 (4) Theme at end of the year, the subject being assigned in 
 the first semester. The candidate must show ability to attack a 
 practical problem and to look up the literature relating to it. 
 If the candidate passes his Probejahr he receives a certificate. 
 He then enters the system immediately, although some prefer 
 to take an extra year as a practice teacher. 
 
 The objections to this system are that accidents play too great 
 a part, and that there is entirely too much emphasis on 
 the study of methods of teaching. Although some study of 
 methods is necessary to give training in the grouping of 
 material, and in getting different points of view, still to the 
 question as to whether pedagogic training is of much aid. the 
 answer " No " is more often given than any other. No inves-
 
 62 Teachers College Record [128 
 
 tigation of the work of the other states is required as in Prussia 
 and other parts of Germany. Only once (1901) was an investi- 
 gation into the schools of the other German states made. A 
 system, definite for all, like the Prussian, would be a good thing 
 for Elsass-Lothringen. 
 
 If we compare the requirements for the high-school teacher 
 of mathematics in our country with those of Elsass-Lothringen, 
 we see that in pure irathematics the requirements for the first 
 degree in that province are about the same as those for college 
 graduation here and for license to teach in our better high 
 schools, except in places like New York where they are slightly 
 higher. We have absolutely no requirement in applied mathe- 
 matics, while the general examination in pedagogy is not at all 
 comparable to the general examination in Elsass-Lothringen, or 
 for that matter in any German province. The requirements for 
 the second degree in Elsass-Lothringen would leave very few 
 eligible teachers in this country. We may conclude, therefore, 
 that one of the reasons why Germany can accomplish better 
 results in mathematics than America, is the fuller and more 
 thorough equipment of her teachers.
 
 CHAPTER IX 
 MATHEMATICS IN GERMAN TECHNICAL SCHOOLS 1 
 
 Donald T. Page 
 
 This article is an attempt to state briefly the main points brought 
 out in a report recently published by Dr. Jahnke of Berlin on 
 the mathematics of the higher industrial schools. 1 The report 
 deals with the special schools of mining, military science, fores- 
 try, agriculture, and commerce. It gives a brief historical state- 
 ment concerning each of the large schools, and a description of 
 the courses in mathematics, with their changes and development. 
 It also treats of the need for mathematics in the requirements 
 for the post and telegraph service, and mentions briefly several 
 institutions that offer courses in higher mathematics. 
 
 The various schools exhibit individual characteristics; but in 
 the ninety years since technical institutions began to develop 
 there has grown up a strong movement against pure mathematics, 
 which has affected the courses in most of the schools. This 
 movement has been caused by a demand for practical problems, 
 and its effect is shown in the new text-books, and in the remod- 
 eling of courses so that the calculus and mechanics are taught 
 in one course under the name " Higher Mathematics and 
 Mechanics." 
 
 Opposition to this movement is felt by many of the best engi- 
 neers who assert that an engineer must have a thorough knowl- 
 edge of mathematics and that the calculus and the theory of equa- 
 tions are especially valuable in developing thinking ability. 
 
 THE SCHOOLS OF MINING 
 
 Most of the schools of mining have the same entrance require- 
 ments as the universities, and some of them have an extra re- 
 
 *Die Mathematik an Hochschulen fur besondere Fachgebiete, von 
 
 Dr. E. Jahnke, Etatsmassigem Professor an der Kgl. Bergakademie, 
 Berlin, Leipzig and Berlin, 1911. 
 129] 63
 
 64 Teachers College Record [130 
 
 quirement of at least one year of practical work. The course 
 is usually three years in length. In Prussia many of the gradu- 
 ates take civil service examinations. The mathematics taught in 
 the Berlin Academy illustrates fairly well the ground covered, 
 and is given in the following table : 
 
 1st year 
 
 2nd year 
 
 3rd year 
 
 Analytic Geometry, 
 Algebra, Descriptive 
 Geometry 
 
 Higher Mathematics 
 and Mechanics 
 
 Theory of Errors and 
 Method of Least 
 Squares 
 
 The use of models and instruments is encouraged, while part 
 of the allotted time is reserved for the solution of practical 
 problems. Freiburg Academy (1765) was one of the first schools 
 to advocate vocational education. The course is four years in 
 length and has not been as much affected by the movement 
 against mathematics as the courses in other schools. 
 
 MILITARY SCHOOLS 
 
 In the military schools there is a growing emphasis upon ap- 
 plied mathematics. A description of the course at the Berlin 
 Military Academy will give a general idea of the mathematics 
 offered in these schools. At Berlin the course is three years 
 in length. The entrance requirements are plane geometry, algebra 
 through quadratic equations, logarithms, and plane trigonometry. 
 The first year's course includes spherical trigonometry, plane 
 analytic geometry, part of the differential calculus, and infinite 
 series. The second year's work completes the differential and 
 integral calculus, and takes solid analytic geometry, some prob- 
 lems in analytic mechanics, and some astronomy. In the third 
 year surveying and more advanced astronomy are taken. 
 
 FORESTRY SCHOOLS 
 
 In some of the forestry schools the course is three years long, 
 in others four. In some, an apprenticeship of several months 
 to a chief forester is required, in addition to the general re- 
 quirement of certificate for entrance to a university. The for- 
 estry schools in South Germany offer more courses in pure
 
 13 1 ] The Present Teaching of Mathematics in Germany 65 
 
 mathematics than do those in North Germany. The mathematics 
 courses generally include the calculus, geodesy, plan drawing, 
 forest computations, and statistics. 
 
 SCHOOLS OF AGRICULTURE 
 
 The schools of agriculture give few courses in mathematics, > 
 usually surveying, levelling, and plan-drawing. 
 
 SCHOOLS OF COMMERCE 
 
 The mathematics in the schools of commerce usually consists 
 of commercial and political arithmetic ; but courses in higher 
 mathematics are occasionally offered. 
 
 POST AND TELEGRAPH SERVICE 
 
 The requirements for entrance to the Post and Telegraph Ser- 
 vice do not include mathematics ; but schools that prepare candi- 
 dates have frequently offered courses in analytic geometry, the 
 calculus, differential equations, and mathematical physics. Higher 
 mathematics has proved helpful to telegraph engineers who have 
 studied it, and is therefore urged as a needed addition to the 
 present requirements. 
 
 Industrial conditions in Germany are such that the need for 
 both pure and applied mathematics is increasing. Mathematicians 
 are far behind in the solution of problems raised in physics and 
 engineering. These conditions seem likely to bring about a 
 strengthening of the present position of mathematics and a weak- 
 ening of the opposition to it.
 
 CHAPTER X 
 THE GERMAN MIDDLE TECHNICAL SCHOOLS 1 
 
 Miriam E. West 
 
 This report concerning mathematics instruction in the German 
 middle technical schools of the machine industry consists of 
 six chapters which treat of the development of these schools, 
 their organization, the mathematics instruction given in them, 
 the text-books, the method of treatment of the different subjects 
 of instruction, and the preparation of the teacher of mathematics. 
 
 Public technical education in the schools of Germany is a 
 product of the nineteenth century. After the invention of the 
 steam engine, when technical instruction was flourishing in 
 France, there came a demand in Germany for technical train- 
 ing. Peter Wilhelm Beuth was the founder of public technical 
 instruction in Prussia. Between 1817 and 1821 four schools 
 were established in Germany, and within a very short time 
 eighteen more. The majority of these schools had a one-year 
 course, but the institution in Berlin, and several others, had two- 
 year courses. The mathematical instruction of the Prussian 
 schools included geometry, arithmetic, algebra through quadratic 
 equations, a little stereometry, and a little trigonometry. In 1850, 
 there were added the principal properties of conies, and theo- 
 retical and practical surveying. In 1870 a reform was instituted. 
 The schools were made three-year schools, and in the two lowest 
 classes general instruction in the languages, history, geography, 
 and the sciences was given. In the third class the instruction 
 was divided into four sections : 
 
 (a) Preparation for entrance to higher technical schools. 
 
 (b) Preparation for the building trade. 
 
 1 Der Mathematische Unterricht an den Deutschen Mittlern Fachschulen 
 der Maschinenindustrie, von Dr. Heinrich Griinbaum, Lehrer der Mathe- 
 mamatik und Physik am Rheinischen Technikum in Bingen. July, 1910. 
 
 66 [132
 
 133] The Present Teaching of Mathematics in Germany 67 
 
 (c) Preparation for technical mechanical work. 
 
 (d) Preparation for technical chemistry. 
 
 The mathematics course was broadened to include the follow- 
 ing subjects: determinants, combinations, binomial theorem, com- 
 putation of logarithms and trigonometric functions by infinite 
 series, continued fractions with applications, spherical trigonome- 
 try, the elements of analytic geometry. The completion of a five- 
 years' course in a Realschule was required for entrance to these 
 schools. A number of Realschulen, at this time, offered this 
 technical education by adding to their five years two more of 
 general instruction and one year of vocational instruction. These 
 became later the Oberrealschulen. This general education with 
 only one year of technical instruction proved unsatisfactory, so 
 in 1889 the Society of German Engineers took the matter in 
 hand. As a result the middle technical schools were established, 
 offering two years of vocational training. Later another half 
 year was added, making five half year classes. They required 
 for entrance, besides the completion of the work of a Realschule, 
 at least two years practice at some vocation. 
 
 There are about forty institutions in Germany which may be 
 classed as middle technical schools. Some of these are state 
 schools; others are private or town schools. Of these the 
 Prussian Maschinenbauschulen are controlled by the state, and 
 the final examinations are given by the royal examination com- 
 mission. For entrance to these schools is required the " Einjahr- 
 igen-Zeugnis " (a certificate which exempts the holder from 
 one year of military service), seven years in a higher school, 
 the completion of the work of a Volkschule and work at a 
 preparatory school, or the passing of an examination in the fol- 
 lowing subjects: German, arithmetic, algebra through quadratic 
 equations and logarithms, plane geometry, trigonometry through 
 the computation of right-angled triangles, and stereometry. The 
 satisfactory completion of the course in these schools qualifies 
 one to enter the following branches of public service : 
 
 (a) State railroad service, as officers, superintendents, en- 
 gineers, etc. 
 
 (b) Marine service. 
 
 (c) Royal-Bureau for the construction of artillery.
 
 68 Teachers College Record [134 
 
 Similar to these Prussian schools are the technical schools of 
 Hamburg, Bremen and Niirnberg and the royal Fachschule of 
 Wurzburg. These are state schools and are included in the 
 forty mentioned above. They have a more elaborate organiza- 
 tion than the Prussian Maschinenbauschulen. Hamburg pre- 
 pares for five distinct vocations: machine construction, elec- 
 trical engineering, ship building, construction of ship machinery, 
 and ship engineering. Some of the other schools have a larger 
 variety of courses, and some fewer. The private schools, since 
 they are not controlled by the state, can adapt themselves more 
 readily to the needs of the various vocations, and many of them 
 keep up with the changes in industry better than the state 
 schools. 
 
 The pupils who enter the middle technical schools may be 
 divided into two groups. Those of the first group possess the 
 " Einjahrigen Zeugnis," have usually completed the six classes 
 of a higher school, and have had one or two years of practice 
 in a workshop. The pupils of the other group possess only a 
 Volkschule education and have had longer practice. Mean- 
 while they have increased their knowledge in industrial mathe- 
 matics by attendance at some continuation school. In some 
 schools the pupils of the second group are put in a preparatory 
 department, while in others the pupils entering are divided into 
 two sections according to their preparation. 
 
 The following is the mathematical curriculum in the Prussian 
 Maschinenbauschulen : 
 
 I. ALGEBRA: powers, roots, logarithms, equations of the first 
 degree, exponential equations, arithmetic and geometric series, 
 interest and annuities, convergence and divergence of infinite 
 series, binomial theorem, exponential and logarithmic series, 
 natural logarithms, maxima and minima, graphical solution of 
 numerical equations. 
 
 II. PLANE GEOMETRY: the most important propositions of 
 elementary geometry, circle, area of plane figures, proportion, 
 construction problems. 
 
 III. TRIGONOMETRY: trigonometric functions and their rela- 
 tions to each other, use of trigonometric tables, computation of 
 triangles, quadrilatrals, and regular polygons.
 
 135] The Present Teaching of Mathematics in Germany 69 
 
 
 
 IV. STEREOMETRY: straight lines and planes in space, the tri- 
 hedral angle, the regular solids; surfaces and volumes of the 
 prism, pyramid, cylinder, cone, sphere, truncated solids, parts 
 of the sphere; general methods for computation of solids, such 
 as Guldin's and Simpson's rules; application to computation of 
 volume and weight. 
 
 Many of the schools have introduced courses in differential 
 and integral calculus. This is especially true of the private 
 schools. 
 
 The aim of instruction in the middle technical schools is dif- 
 ferent from that in the higher or so-called cultural schools. 
 In the former mathematics is studied from a practical stand- 
 point, and as a tool for solving the technical problems of the 
 various vocations. It is, therefore, impossible to borrow the 
 methods of the higher schools, in which mathematics is usually 
 studied as an end in itself. A hundred years of vocational 
 schools has not been sufficient to create methods peculiarly 
 adapted to technical instruction. The text-books used in the 
 two schools are largely the same. The fact that the instruction 
 is similar in the two schools is due partly to the fact that many 
 of the vocational schools were originally organized in connec- 
 tion with higher schools. The pupils in these schools are older 
 than those in the higher schools, being from eighteen to twenty- 
 two years of age. 
 
 The instruction in the technical schools has or should have 
 certain distinguishing features. The problems in algebra, so Dr. 
 Griinbaum asserts, should be chiefly practical ones, for which 
 geometry, mechanics, and physics offer a good field. In order 
 to obtain the required skill it is necessary for the pupil to work 
 many problems, but it is of no great value for him to work those 
 for which he will later have tables. Practice in use of the slide 
 rule and shortened methods of computation as well as practice 
 in drawing is considered important. 
 
 In the teaching of mathematics there is danger of too much 
 emphasis being placed upon the use of formulas. If mathe- 
 matical principles are dried up to formulas and the only mental 
 work consists in applying them to certain particular examples, 
 the natural result is that the principles behind the formulas are
 
 7O Teachers College Record [136 
 
 soon forgotten, and the pupil has simply a collection of rules, 
 with no knowledge of their application. It is important to keep 
 the principles in mind. The pupil need not be expected to re- 
 member a large body of formulas. The more simple ones he 
 can derive easily for himself, and for the more difficult ones 
 which have been previously derived in class he may have access 
 to a book containing a collection of them. 
 
 In algebra the most important thing for the pupil to acquire 
 in the first few weeks is skill in solving equations. The other 
 elementary instruction in algebra should be made of service to 
 this fundamental problem. The fundamental operations, factor- 
 ing, and fractions should be taught with this in view. On this 
 skill will depend ease in development of formulas. One notices 
 the dependence of these schools on the higher schools in the 
 use of the old traditional problems, the boy with the nuts, the 
 farmer's wife with the eggs, the tank with the pipes, etc. The 
 real technical problems are seldom found in the text-books. The 
 treatment of powers, roots, and logarithms is pronounced by all 
 to be too formal. There is too much dependence on the rules 
 concerning exponents and indices. 
 
 In many of the text-books great stress is placed upon the 
 solution of the radical equation. Yet the text-book writer often 
 fails to notice that values of x obtained do not satisfy the equa- 
 tions. For example: 
 
 V 3 6 + x 
 36 + x = 324 + 36 V x -f- x 
 
 V~x" = 8 
 
 A graphic representation of the parabolas y 2 = 36 + x and 
 (y i8) 2 = x would make clear the difficulty in this solution. 
 
 The function concept has penetrated very little into the in- 
 struction. Such concepts as those of constant, dependent and 
 independent variable, and function are of great value in the 
 solution of equations and in the use of formulas. There is still 
 a large space in the text-book devoted to the old style propor- 
 tion. If this were put in the field of the function concept, it
 
 137] The Present Teaching of Mathematics in Germany 71 
 
 would be fruitful. For proportion only y = ax is needed, and 
 not the form y x : y 2 = x t : x 2 . 
 
 In the solution of equations of higher degree an average 
 course would give the remainder theorem and the theorem con- 
 cerning the relationship between the coefficients and roots of an 
 equation of the nth degree. This gives a method for finding 
 the integral roots. The most common solution of the equation 
 f (x) = o is by finding the intersection of its curve with the 
 x = axis or by the intersection of the two curves 
 
 y = m (x), 
 and y = n (x), 
 if m(x) n(x) f (x). 
 
 Graphical methods are used for the solution of cubic and bi- 
 quadratic equations. There is little practical need for the solu- 
 tion of equations of higher degrees, difficult quadratic equations 
 with several unknown quantities, or reciprocal equations. 
 
 In geometry, the Euclidean method of proof is used very little. 
 Symmetry is an important means of proof. The propositions 
 concerning parallel lines are proved through motion. In all the 
 new books many of the construction problems are giving way 
 to problems in computation of figures. The introduction of the 
 ideas of kinematics into plane geometry is to be commended. 
 Through these should come the explanation of the ideas of trans- 
 lation, rotation, rolling, and the simple propositions concerning 
 the moment of force. The study of solid geometry in connection 
 with plane geometry is of advantage tq the pupil in the tech- 
 nical school, for he has as much need of the former as of the 
 latter, and, by connecting them, he is able to transfer the ideas 
 of plane geometry to the figures in space. The instruction in 
 stereometry should not be simply the application of formulas 
 to problems in computation of surfaces, areas, and volumes, but 
 should serve above all to strengthen the idea of space. More- 
 over it should prepare for possible applications. This can be 
 attained if the pupil not only computes the area and volume 
 of the regular solids but is accustomed to compute other solids 
 by breaking them up by means of planes into elementary bodies. 
 
 In plane trigonometry the instruction is similar to that in the 
 higher schools, but many are of the opinion that such thorough
 
 72 Teachers College Record [138 
 
 treatment is unnecessary. All practical need" would be satisfied 
 if a detailed handling of the right angle triangle and its applica- 
 tions, together with the sine and cosine propositions, were given. 
 A review of the text-books shows that mechanics and physics 
 are rich in examples which require the solution of the right 
 angle triangle. But for the scalene triangle it is with difficulty 
 that the authors find practical problems. Great exactness is not 
 necessary in practical work, so logarithmic solutions are not im- 
 portant. 
 
 In analytic geometry the function which is of the greatest im- 
 portance is y = ax 2 -f- bx .+ c. Of the higher curves, y = ax, the 
 entire function of the nth degree, the exponential curve, the 
 
 2 TT 
 
 curves y = sin x and y = A sin (x 6), the cycloid and 
 
 several spirals (in polar coordinates), are the principal ones 
 considered. The solid analytic geometry is based on the plane 
 analytic geometry. The direction cosines of a line and the 
 geometric significance of f (x, y, z)=o should be familiar 
 to the pupils. The pupils comprehend easily and with interest 
 those equations of surfaces which are closely connected with 
 the equations of well-known curves, as: 
 
 a b 
 
 x 2 , y 2 , /z'\ 
 
 I = i 
 
 a 2 b 1 \c 2 / 
 
 x 2 +y 2 + (z 2 ) =r 2 , etc. 
 
 The fact that differential and integral calculus has not been 
 introduced into the schools of Prussia is due mainly to one man, 
 Professor Gustav Holzmuller, who has persistently fought its 
 introduction. He would have the problems which would natur- 
 ally be solved by calculus solved by the so-called elementary 
 method. In his text-book, " Ingenieur Mathematik," he has 
 the following: 
 
 Let the equation of the curve be 
 
 i 2 " 3 4
 
 139] The Present Teaching of Mathematics in Germany 73 
 
 To find the tangent to this curve which makes a certain 
 angle a with the y axis, the following formula is used: 
 
 tan a = a + by + cy 2 + dy 8 . 
 Likewise given the equation 
 
 q = a + by -f- cy 2 + dy 3 , 
 
 q being the slope of the tangent to the curve at any point, the 
 equation of the curve is 
 
 x= ^y+by! + sz! +dy' 
 
 1 .3 4 
 
 How does this differ from differentiation and integration? In 
 
 the same elementary manner he develops the general binomial 
 theorem, exponential and logarithmic series, the sine and cosine 
 series. For the summation of the integral [ x" dx it is neces- 
 sary, if this method is used, to know the sum of the powers 
 
 i n + 2 n + 3 n . . . m n = 2 x". 
 
 i 
 
 These can be found by this identity 
 
 (a+i) n = a n +na n t 1 + 
 
 putting a = o, i, 2, m and adding all the equations. 
 
 If this is continued for n = 2, 3, 4, , the formula is 
 
 obtained 2 x 2 , 2 x 3 , 2 x 4 , etc. Contrast this with the general 
 interpretation of 2 x" by integration. In a similar manner for- 
 mulas are obtained for the areas and volumes of solids. These 
 problems, which until recently have been treated by this elemen- 
 tary method, are now coming to be handled by the method of 
 the calculus, which is much simpler and has fewer formulas. 
 
 The time given to mathematics need not be increased for the 
 introduction of the calculus, since the calculus merely affords a 
 new method for solving old problems. After the idea of differ- 
 ential quotients is made clear and the simple functions and their 
 combinations are differentiated, then follows the discussion of 
 curves, maxima and minima, turning points, slope, rate, accelera- 
 tion, and infinite series. After an introduction to integration the 
 entire remaining time can be given to the important technical 
 computations, (surfaces, solids, moments of inertia, centers of 
 gravity.) Books are now being published which strive to supply 
 the need of the technical schools in this direction.
 
 74 Teachers College Record [140 
 
 Up to the present time the required preparation for a teacher 
 in a middle technical school has been the graduation from a nine- 
 class school, together with four years academic training. It is 
 necessary that the teacher have a broad view of the need of 
 mathematics in the technical callings. The question which as 
 yet has not been settled is, where can the teacher obtain this 
 broad view? If he goes to the University he will not get it, 
 although several courses are now being offered at Gottingen 
 which help to meet this need, such as general applied mathe- 
 matics, mathematical instruction in higher schools, technical 
 mechanics. It would seem as though the technical high 
 schools would offer the necessary preparation for the teacher. 
 But if he is trained here, he is obliged to specialize along one 
 line and thus he fails to get a view of the needs of other lines. 
 The technical high schools cannot afford to offer special courses 
 for teachers, since the demand in this line is so small, only 
 about six being needed each year. The training is still an un- 
 settled problem. The engineer lacks the necessary training for 
 a teacher, and the teacher trained in pure mathematics lacks the 
 necessary knowledge of mathematics as applied to mechanics.
 
 CHAPTER XI 
 
 MATHEMATICS IN GERMAN SCHOOLS OF 
 NAVIGATION 1 
 
 Donald T. Page 
 
 In early times there was little need for schools of navigation. 
 The young sailors could learn the most important requirements 
 by experience on board ship, and could learn from the older 
 sailors what few scientific principles were necessary. The leisure 
 time on a voyage and during the winter season was employed 
 to advantage in such studies. 
 
 As commerce grew, ships of larger size were built, voyages 
 became longer, and better trained seamen were needed. During 
 the eighteenth century this need of trained seamen was felt in 
 the trade with the East Indies and in the great growth of trade 
 at the close of the War for Independence in America. To satisfy 
 the demand, private schools sprang up. 
 
 Each private school had its own standards of excellence. In 
 many cases the student gave his teacher a certificate of com- 
 mendation in exchange for his certificate of graduation. The 
 school at Hamburg rendered valuable service for many years by 
 publishing a nautical almanac. 
 
 In 1793 Emden made the requirement that candidates for 
 positions of mate or captain must pass an examination. The 
 other states soon made similar requirements; but certificates of 
 one state were not accepted in another state, and there was no 
 uniformity in the requirements until 1870, when the Bundesrat 
 instituted the present system. 
 
 After the examinations were made compulsory, public schools 
 of navigation were finally organized. There were special schools 
 established for the education of captains of small craft, and six 
 
 *Der Mathematische Unterricht an den Deutschen Navigationsschulen, 
 von Dr. C. Schilling und Dr. H. Melddau, Leipzig and Berlin, 1911. 
 
 141] 75
 
 76 Teachers College Record [142 
 
 such schools are now in existence. These also prepare for 
 entrance to the higher schools with which this article is chiefly 
 concerned. 
 
 There are eighteen of these higher schools offering courses 
 leading to examinations which qualify for the position of mate 
 or captain of large ships. There are no entrance requirements. 
 The purpose of the course is to prepare for examination, and 
 for this reason the subject matter and methods of study are 
 somewhat limited. The course for mates is about eight months, 
 and for captains about four months ; but at least two years' ser- 
 vice must intervene. 
 
 Examinations are in charge of an examination commission at 
 each school, but are provided for in such a way as to give 
 uniformity throughout the country. To be eligible for the mate's 
 examination, a candidate must have had at least forty-five months' 
 experience after reaching fifteen years of age. This experience 
 must include twenty-four months' service as able seaman on a 
 merchant vessel and twelve months on a sailing vessel. For the 
 captain's examination, a candidate must have served twenty-four 
 months as mate, and must present notes and computations show- 
 ing that he has had practice in taking nautical observations. 
 These requirements bring the age of candidates up to 22 years 
 for mates and 26 years for captains. During a recent year 
 there were examined 746 of the former and 441 of the latter. 
 
 The examination consists of three parts, oral, practice, and 
 written. In order to preserve uniformity the most weight is 
 given to the written part, which consists of sets of problems 
 divided into three groups of seven subjects. Out of a supply 
 of problems, packets are made up so that each candidate re- 
 ceives one problem on each of the twenty-one subjects. The 
 time for solution occupies from seven to twenty hours. 
 
 FIRST GROUP SECOND GROUP THIRD GROUP 
 
 Use of charts Compass German 
 
 Computation of altitude True course Algebra 
 
 Longitude and time Latitude Planimetry 
 
 Lunar distances Latitude from stars Stereometry 
 
 Position by two altitudes Position by dead reckoning Plane trigonometry 
 
 Variation of compass Winds and currents Spherical trigonometry 
 
 Signals Log book Physics
 
 143 j The Present Teaching of Mathematics in Germany 77 
 
 In the first group all problems must be correct; but four out 
 of each of the other groups are sufficient. 
 
 The amount of pure mathematics needed is merely algebra, 
 geometry, and trigonometry; but students entering a school of 
 navigation have been out of school so long that they require 
 much review of elementary principles. J'hey have also devel- 
 oped habits of concrete thought which make abstract concep- 
 tions hard for them to grasp; therefore the work is limited 
 to mere preparation for the examinations. Then, too, the 
 examinations themselves have followed a stereotyped plan so 
 that the candidate knows just what sort of problems to expect. 
 This also prevents any broadening of methods in the mathematics 
 work. Some changes in the requirements or methods of the 
 examinations might well be instituted in order to give a broader 
 training. The requests for post-graduate courses, made by many 
 who feel their deficiencies, and the training-ships of the North 
 German Lloyd are influences in the right direction. 
 
 The method of selecting teachers also needs to be changed. 
 At present there are two classes of teachers, those who have 
 much nautical experience but little education, and those who 
 have a thorough education but little nautical experience. A 
 course should be arranged so that a prospective teacher may 
 acquire both education and experience. Then, too, the system 
 of appointment and promotion is such that the teachers in these 
 higher schools of navigation are more than sixty years old. This 
 gives an undue conservatism to their influence. 
 
 The system of examinations has been very beneficial in bring- 
 ing seamanship up to a high standard, and the schools of navi- 
 gation have done well in handling the material which comes to 
 them ; but many changes will soon be necessary. Modern inven- 
 tions are making navigation into a new science and removing 
 the need for many of the former methods. The schools should 
 keep abreast of this progress.
 
 CHAPTER XII 
 
 COMMERCIAL PROBLEMS IN THE HIGHER SCHOOLS 
 OF GERMANY 
 
 W. F. Enteman 
 
 In this review of a report on the " Commercial Problems in 
 the Mathematical Instruction of the Higher Schools," 1 will be 
 found, as in the original publication, (a) some discussion of the 
 field of commercial problems, (b) a brief review of text-books, 
 both old and new, (c) a discussion of the important aids in 
 teaching the subject, and (d) some general conclusions. One 
 of the aims of the report is to call attention to the cause for 
 the existence of commercial problems in the mathematical in- 
 struction in Germany. 
 
 The German Minister of Education has said, " Not book 
 knowledge but understanding of life is desired." Shall the 
 schools tackle the question? Those who believe the object of 
 schools is the education of young men for citizenship answer in 
 the affirmative ; those who believe the school is for the intellectual 
 only answer in the negative. In the opinion of Dr. Timerding 
 commercial problems have a meaning from either standpoint, 
 and in support of this view he makes the following statements. 
 
 (1) The calculations of practical problems have the same 
 psychological value as the calculations of long tedious problems 
 that have no bearing on life. 
 
 (2) Proofs for the mathematical formulas needed develop the 
 formal side. 
 
 (3) Mathematics is as valuable considered as a useful tool as 
 when it is looked upon merely as an intellectual product. 
 
 (4) Satisfaction is derived from the knowledge of the adapta- 
 bility of abstract formulas. 
 
 *Die Kaufmannischen Aufgaben im Mathematischen Unterricht der 
 Hoheren Schulen, von Dr. H. E. Timerding, Leipzig und Berlin, 1911. 
 78 [144
 
 145] The Present Teaching of Mathematics in Germany 79 
 
 There is no definite standard imposed on the text-books as to 
 the amount of work to be devoted to commercial problems 
 or as to the different topics to be treated. The personality of 
 the teacher is the important factor in determining this amount. 
 He can give flesh and blood to the framework by securing local 
 problems of vital interest, or he can make flimsy the work of 
 the best text-book. In arranging a course of study, a scientific 
 method should be followed and not a whimsical one. It is not 
 a simple matter to correlate the theory and practical applications 
 of any science in teaching, and this is especially true with regard 
 to mathematical instruction. Commercial arithmetic, for exam- 
 ple, must be treated with theoretical arithmetic as heretofore, 
 and as a separate subject, but a far more definite method must 
 be worked out. 
 
 The condition of the work rather than the manner of instruc- 
 tion is the main topic considered in this report and the following 
 points are emphasized: 
 
 (1) That it is possible to make a collection of scientific prob- 
 lems based upon practical knowledge. 
 
 (2) That the empty formalism of problems must be discour- 
 aged. 
 
 REVIEW OF THE FIELD OF COMMERCIAL PROBLEMS 
 
 For a general view of the field from which problems may 
 be drawn we must consider the forms of our commercial activity 
 in general. There are two of these general forms: (i) Traffic 
 in Goods, and (2) Traffic in Money. The tradesman is espe- 
 cially concerned with the first, the banker with the second. The 
 tradesman must be able to find a suitable sale price for an article 
 when the cost price is given, or, if he has a fixed sale price, he 
 must find a suitable cost price in order that he may make a 
 certain profit. The subject of alligation is applicable to problems 
 in this first field. It is taught in the general schools, and usually 
 in connection with linear equations. It treats of the proportions 
 in which goods can be mixed for pleasant taste or for cheap- 
 ness, as, for instance, different grades of tea, coffee, wine, etc. 
 The fundamental principle is simple; namely, that the total cost
 
 8o Teachers College Record [146 
 
 of a quantity of mixture must equal the cost of its separate con- 
 stituents. The formula 
 
 m 1 p t + m 2 p 2 + m 8 p 3 +. . . . 
 
 m t + m 2 4- m s +. . . . 
 
 in which m t , m 2 , m 3 , .... stand for amounts, and p lf p 2 , 
 p 3 , . . . . stand for price, can be used if the price of a 
 mixture is to be found when the price and amounts of its con- 
 stituents are given; or if there are only two constituents to be 
 used, the quantity of one of them being given to find the quan- 
 tity of other that will give a mixture worth a certain price. For 
 an illustration see Feller and Odermann, " Das Ganze der kauf- 
 mannischen Arithmetik," Leipzig, 1908. Another use for alliga- 
 tion is in the solution of problems on the coinage of money. It 
 may be used to find the value of a known mixture of metal of 
 which certain coins are composed, or the weight of a certain kind 
 of precious metal which is to be mixed with a known weight of 
 another metal to obtain a required mixture. Also it can be used 
 to find the proportion of alloy and metal for the standard coins, 
 or to find the real value of money when the constituent parts 
 are given. This is treated rather superficially in the general 
 schools. 1 
 
 The calculations which have to do with money are in general 
 of greater worth and interest to the schools than any others. 
 We can divide this subject into two parts : Primary Calculation, 
 as in the case of simple interest; Higher Calculation, as in the 
 case of compound interest and the more advanced work in simple 
 interest. 
 
 Under Primary Calculation may be placed: 
 
 (a) Percentage, with the important formula (i) p = br. 
 
 (b) Interest, the fundamental formula of which is (2) 1 = 
 prt. 
 
 For convenience in account current the interest rate is reduced 
 to a rate per day. If t lf t,, tg . . . indicates the time in days, 
 and a lf a 2 , a s ..... the deposits which lie in the bank up 
 
 1 The extensive use of alligation in Germany is one of the surprises 
 that greet the American student. There is at present a little effort 
 to revive the study in the United States, but its value seems very slight 
 and the chance of its introduction in the schools is rather remote.
 
 147] The Present Teaching of Mathematics in Germany 81 
 
 to the time when the account is closed, we have the following 
 formula : 
 
 (3) I=(a 1 t 1 + a 2 t, + a 3 t 3 + ....... )q, in which q = 
 
 . This method of calculating interest is called the progressive 
 
 method. If the accounts are closed every half year, the follow- 
 ing formula is used : 
 
 (4) 1= [(a 1 + a z + a, . . . .)i8o a^iSo t t ) a 2 (i8o- 
 
 The method of calculating interest by this formula is known 
 as the retrograde method. Another formula often used in de- 
 termining interest is: 
 
 (5) I=[a 1 (t 1 -t t ) + (a 1 + a f )(t -t.) + (a 1 + 
 
 (c) Discount. 
 
 (6) A = a(i + rt). This is called the Hoffmann discount 
 formula (1731). 
 
 (7) a' = A (i rt). This is called the Carpzo discount 
 formula (1734). These two correspond to our formulas for true 
 discount and bank discount. In 7, a' is an approximate value 
 for a when rt is a small fraction. 
 
 (d) Equation of Payments. 
 
 at -4 Q t* I 3 f" I 
 1 Lf | el., L.> ] do Lo ] * 
 
 A L mm o o 
 
 (8) t=r ~rTT 
 
 aj -f a 2 ~r a 3 T- 
 
 (e) Partnership. This has to deal with the distribution of a 
 certain profit among a given number of partners according to 
 a given ratio. If A represents the amount that is to be divided, 
 
 Ci,c 2i c 5 the given ratio, and A 1 ,A 2 ,A 3 .... the 
 
 amount that each should receive, we have the formula 
 
 CjA c 2 A 
 
 c t + c 2 -f- . . . c, + c 2 + . . . . 
 
 Under Higher Calculation may be placed: 
 (a) Compound interest, the fundamental formula of which is 
 
 (10) A = p(i +r)*. 
 
 The next formula is especially interesting when n becomes 
 indefinitely large and the interest is constantly due and constantly 
 added to the capital. Here we may use instead of 
 
 (11) A-
 
 82 Teachers College Record [148 
 
 If this is compared with (10) we have 
 
 r' = nat log (i + r), 
 which may be developed into the series 
 
 r T* T* . 
 
 r' = r I-- + h ]. 
 
 234 
 This last formula was introduced by Jacob Bernoulli in 1690. 
 
 (b) Calculation of Income or Revenue. The payment of a 
 debt through annual payments and the fixing of allotment for 
 the payment of a loan may be put under this subject, and the 
 following formula for solving such problems is given. 
 
 (12) S = s(i+q + q 2 + q 8 4 J , +q t - 1 ), 
 
 where q = i + r. This formula is usually written as follows : 
 
 which is a familiar formula in geometric progression. 
 
 From the calculation of final S of an income follows directly 
 the calculation of the cash value B, we have 
 
 q* (q - 1) 
 
 We can also find the natural earnings or increment if we assume, 
 instead of separate income payments which follow at regular 
 intervals, a continuous payment. There falls by this process 
 upon the time element dt an infinitely small value 8 dt and the 
 cash value of income will, if payment is for the period t to t lf 
 be expressed by the integral 
 
 (15) B= 
 
 J 
 
 Instead of a constant stream of income, a variable one may be 
 used in which S is a function of the time and we must write 
 S(t). Then we have 
 
 (16) B= e-^t) dt 
 (c) Insurance. 
 
 REVIEW OF TEXT-BOOKS . 
 
 It is a great task to review all the works on commercial arith- 
 metic, since every arithmetic takes up at least something on the 
 subject. Only a few of the most important works can be men- 
 tioned at this time.
 
 149] The Present Teaching of Mathematics in Germany 83 
 
 Martus (1903) "Mathematical Problems for the Upper 
 Classes." Besides interest and partnership problems, there are 
 other problems which deal with time in relation to money cal- 
 culation. As an illustration of a problem with an incorrect 
 result the following may be given from this book : 
 
 How much must a man 30 years of age deposit in a bank 
 annually until he is 65 years of age in order that he may receive 
 3,000 Marks yearly for 10 years, if 3 l /2% compound interest is 
 allowed? The result given is 393 Marks, but no consideration 
 is given to the fact that the man may die before he is 75 years 
 of age. The correct result is 162 Marks. " Why permit a 
 problem with an incorrect result ? " is the comment of the author. 
 
 Bardy's "Arithmetic" (new edition, 1910). This gives real 
 problems only in compound interest and calculations on income, 
 and rather more work of this kind than necessary. 
 
 Miiller and Kutnewsky have a book with complete tables in 
 insurance. This enables the teacher to choose problems to suit 
 himself and the needs of the class. 
 
 A. Schiilke (1906) has a text that covers many topics. The 
 applications of the commercial problems are very accurate 
 throughout. Life insurance problems are given and a general 
 formula is stated, but not a single example under this formula 
 is given. Lottery problems are treated very fully, a feature that 
 is practical in Europe although happily obsolete here. In spite 
 of some defects, this seems to be the best book examined. The 
 problems are taken from real life. 
 
 Schulz and Pahl (1906) are the authors of a book which is 
 in many respects the opposite of the one just reviewed. Com- 
 pound interest and investment problems are very few. Mor- 
 tality problems and savings bank problems are given, but no life 
 insurance problems appear. 
 
 A Bavarian text-book by Hoffman (1892) has much that is 
 commendable. Life insurance is treated briefly but skillfully. 
 
 In general these new books make an effort to obtain problems 
 from the experience of the citizens and to place before the 
 student statements that will give him sound and rational notions 
 on the subject. It is difficult to find texts to supplement the 
 work in commercial arithmetic. Feller and Odermann have a
 
 84 Teachers College Record [150 
 
 book that is well written but it is not one that can be given to 
 the pupils. 
 
 Thaer and Rouwolf (1911) treat commercial problems in an 
 abbreviated form. 
 
 Moritz Cantor's " Political Arithmetic " is a small book, but 
 a very good one. Dr. Timerding makes the following comment 
 on this work : " In the work of selecting problems a word of 
 caution may be necessary. There is danger that this attempt 
 to select interesting problems for the student may be carried 
 too far. Cantor is whimsical, being inclined to skip about, and 
 may not always want what is best." 
 
 The so-called story problems fall into two classes: (i) real; 
 (2) unreal or fanciful. If only formal training is desired, the 
 latter class of problems with their usual rules is successful with 
 young minds. If, on the other hand, immediate practical instruc- 
 tion is the aim, real problems are necessary. In the old books 
 the unreal problems were separated from the others. Some of 
 these problems which appear to-day in our texts are of great age ; 
 and the origin of many can be traced to Egypt, India, Greece 
 and China. The following is an example of an old Egyptian 
 problem. How many pigeons are on a ten-round ladder, if there 
 is one on the first round, two on the second, four on the third, 
 etc.? The charm is in the result that there would be 512 pigeons 
 on the last round. 
 
 Some of the fairy problems are so old that no one knows 
 where they came from. The following was taken from a book 
 which appeared in 1494: On the top of a tree, which is 60 ells 
 high, sits a mouse and on the ground beneath the tree is a cat. 
 The mouse climbs down ell each day and climbs up ell each 
 night. The cat climbs up i ell each day and climbs down ell 
 each night. The tree grows ^ ell each day and shrinks ell 
 each night. When will the cat reach the mouse and how high 
 will the tree be at that time? 
 
 In the works of Paciuolo (1494) we first find interest cal- 
 culation. 1 The calculation of interest by means of the linear 
 equation is given in a work of the Renaissance period, and in 
 1600 appears in an arithmetic. Stevin in 1582 gave an interest 
 
 1 This is not true, for such work appears in a number of books pub- 
 lished before 1494.
 
 151] The Present Teaching of Mathematics in Germany 85 
 
 table ; Jost Biirgi introduced some calculations with decimal frac- 
 tions j 1 Insurance appeared during the seventeenth century ; Hal- 
 ley, in 1693, for the first time found the present worth of an 
 annuity based upon mortality tables. The leading features of 
 Commercial Arithmetic began to develop in the eighteenth cen- 
 tury. Halcke, in 1719, published his " Sweet Meat Thoughts," 
 much of the work being in verse. Many works appeared at this 
 time, and among the number may be mentioned those of Claus- 
 berg (1732) and Biisch (1769). 
 
 About the beginning of the nineteenth century the subject was 
 taken up in the higher schools and the cultural side was empha- 
 sized. Lately the idea of money making has brought the 
 practical side forward. The formal side has much worth, the 
 advantage of which may be seen if we view it from problems 
 which are not merely fanciful. It seems wrong to propose ques- 
 tions which in practice are meaningless. The writer of this 
 report says that he is not opposed to the formal character of 
 problems if the aim of each problem can be made clear. A good 
 example of the above type is one from Euler which results in an 
 equation of the third degree with three positive, integral roots, 
 namely 7, 8 and 10. We can understand the problem better if we 
 write the equation in the following form: 
 
 (x-7) (x-8) (x-io)-o, 
 
 y^ 
 
 or (8240 + 4Ox 2 ) i ox 2 = 224. 
 100 
 
 The problem given is this : There were x persons engaged in 
 a certain business and to their joint capital of 8,240 Marks each 
 added 4Ox Marks. Their profits amounted to x per cent, and of 
 this profit each took iox Marks and then there remained 224 
 Marks. It is required to find x. The interest in this problem 
 lies in the fact that there are three solutions. 
 
 Another good example, if considered from the formula stand- 
 point, is the following: Three brothers inherit 45,500 Marks. 
 The one that received the least amount gave -fa, the one that re- 
 ceived the most gave , the third gave to a charitable institu- 
 tion, after which they all had the same. How much did each 
 
 'This is a superficial statement, not to be taken seriously.
 
 86 Teachers College Record [152 
 
 receive ? The problem sets forth a generosity not found in com- 
 mon life. 
 
 The following problem from Schiilke is an excellent one: 
 Two persons take out life insurance. Each pays 100 Marks 
 yearly, one for 30 years and the other for 10 years. Would it 
 be the same for the company if each paid for 20 years? 
 
 In compound interest, neither the time nor the rate is ever 
 unknown, nor is it practical to count the interest for more than 
 10 years or for such rates as 3$% or 
 
 AIDS 
 
 The principal thing to be emphasized in the treatment of 
 commercial problems is simple numerical calculation. Calcula- 
 tion is the backbone of the arithmetic course of study just as 
 proof is that of geometry. The defective preparation of pupils 
 in the higher schools in practical calculation is a decided fault. 
 Some of the things which might be mentioned as an aid in the 
 presentation of commercial problems are : 
 
 (1) Graphic representation. Often the meaning of a problem 
 as well as the meaning of the result can be made clear by a 
 drawing. Some of the means of representation are very old. 
 From the works of Leonardo of Pisa we find a problem involv- 
 ing proportion, the solution of which is represented by lines. 
 
 (2) Slide rule. Next to graphic aids the slide rule is of 
 greatest value to the engineer. It is not generally used in con- 
 nection with commercial problems. 
 
 (3) Logarithmic tables and logarithmic curves. 
 
 (4) Formula. The formula is perhaps rated too high. It is 
 not of so much importance in commercial as in technical arith- 
 metic. 
 
 (5) Simple illustrations to explain the solution of a problem. 
 The following taken from Findeisen-Clausen's " Examples and 
 Problems for Instruction in Commercial Calculations " (Leipzig, 
 1905) will explain the above. If we have a problem in which 
 we are to find the amount of a note due at some future time 
 and also to change from one standard to another, we might use 
 a river as an illustration. The two banks will correspond to the
 
 The Present Teaching of Mathematics in Germany 87 
 
 two standards. The flowing of the water will correspond to the 
 passing of the time. The crossing of the river corresponds to 
 the changing from one standard to the other. As there are two 
 ways of going from a point on one side of the river to a point 
 farther down the river and on the opposite side, so there are two 
 ways of solving the given problem. A person can go directly 
 across the river and then go to the other point or he may go 
 down the river to a point directly opposite the other point and 
 then cross. Likewise in solving the problem, we may change 
 from one standard to the other and then calculate the amount, 
 or we may calculate the amount first and then change from the 
 given standard to the other standard. Also the formulas re- 
 lating to mixtures and alloys are analogous to the one relating 
 to the center of gravity. The separate articles correspond to 
 the different points, and the different prices to the different dis- 
 tances. Equation of payments could also be used with the 
 formula for the center of gravity. The different amounts cor- 
 respond to the different points, and the periods of time to the 
 distances. 
 
 (6) Graphs. The graphic method leads only to an approxi- 
 mate result, but solutions involving several decimal places 
 have no place in reality. The graph has already won recogni- 
 tion in the schools, and if millimeter paper is used, sufficiently 
 accurate results can be obtained. We might distinguish two dif- 
 ferent methods. 
 
 (a) Graphic statistics. In this case we construct a curve to 
 show at a glance the general inference to be drawn from a set of 
 statistics. As an illustration we could have a curve showing the 
 number of people out of 1,000, say, that die at the different 
 ages. This curve starts at a considerable distance from the 
 abscissa and very irregularly approaches the abscissa which it 
 finally reaches. 
 
 (b) Graphic calculation. If we have given a curve constructed 
 from statistics, as in the example under (a), or from a tabula- 
 tion of the results of experiments, we can obtain the value for 
 any abscissa directly from the curve without referring back to 
 the tables, and can easily interpolate values not given in the
 
 88 Teachers College Record [154 
 
 tables. A graphic table may be used in connection with arbi- 
 trated exchange. It may be shown easily and quickly whether 
 it would be better to send directly to another city or to send 
 through a third place. 
 
 CONCLUSION 
 
 We have seen that there are two opposing views in regard 
 to the study of commercial subjects. The advocates of the 
 one view demand the consideration of practical claims and 
 simple correctness of the problems, with everything in harmony 
 with actual business transactions. The others claim that it is 
 wrong to emphasize excessively business practice, as against 
 theory, and to permit this to interfere with a scientific treatment 
 of the subject. 
 
 We should not endeavor to make the pupil an accomplished 
 merchant; he should only be expected to acquire some knowl- 
 edge of the fundamental principles of commercial transactions. 
 It is not necessary that he should know all of the technical terms 
 and all of the finer details connected with commercial calcula- 
 tions. We should seek to impart only a sound practical knowl- 
 edge through simple problems which are suitable from the stand- 
 point of the pupil. 
 
 The real problems should be separated from the unreal so 
 that the pupil may know which kind he is working with. The 
 aim of unreal problems is to arouse the play instinct of the 
 young mind and thus to animate his study of mathematics and 
 in this they play an important part. The result at which they 
 aim is formal. The pupil learns from amusing and interest- 
 ing problems a new mathematical process or finds a new ap- 
 plication for a process of calculation already learned. On the 
 other hand, the real problem leads him to a simple problem full 
 of meaning. It gives him a moment of real life. It shows him 
 how the things he learns in school can be applied to things out- 
 side. It is placing this directly before him that binds school 
 and life together. 
 
 From this report, we might draw the following conclusions: 
 
 (i) There is no definite course of study in commercial arith- 
 metic in the higher schools of Germany.
 
 155] The Present Teaching of Mathematics in Germany 89 
 
 (2) The subject of alligation is considered to be important, 
 although in this report only the simpler cases were mentioned. 
 
 (3) The formula is used in the solution of problems to a 
 much greater extent than is done in this country. 
 
 (4) The theory underlying certain subjects, as life insurance, 
 theory of probabilities, etc., is considered much more fully than 
 in this country. 
 
 (5) The graph is used to a great extent. 
 
 (6) The question of real problems is with them an unsettled 
 one just as it is with us. On the whole, it may be said that 
 the work that we in the United States are doing in commercial 
 arithmetic compares very favorably with the work being done 
 at present in Germany. At any rate, we seem to have fewer 
 problems that are unreal while pretending to represent modern 
 conditions.
 
 CHAPTER XIII 
 MATHEMATICS IN THE TEXT-BOOKS ON PHYSICS 
 
 Arthur T. French 
 
 This article contains a brief digest of a much more extended 
 discussion under the same title 1 in one of the reports of the 
 International Mathematics Commission. 
 
 Although Dr. Timerding has written in the interests of mathe- 
 matical instruction, it has not been his intention to emphasize 
 mathematics unduly or to prescribe rules for the teaching of 
 physics. He does, however, aim to show that mathematical 
 problems arise in the study of physics and that the solution of 
 these problems is necessary for an understanding of the physics. 
 Many of the difficulties of physics will be made easy through 
 a reform in the teaching of mathematics. 
 
 Physics instruction ought to be concrete and pupils should 
 perform their own experiments. In the lower grades this in- 
 struction should be given without any mathematics. The in- 
 troduction of mathematics too early in the grade lessens the 
 interest in the subject; but before beginning the formal study 
 of physics the pupil should have a fair mastery of all neces- 
 sary mathematics. How much that includes is a difficult ques- 
 tion to answer, but in the beginning, mathematics and physics 
 should be independent. In the majority of schools, instruc- 
 tion in mathematics and instruction in physics go side by 
 side, consequently the pupil cannot use his knowledge of 
 the one subject in the study of the other. This arrangement 
 is not necessary since one can just as well be given be- 
 fore the other. Mathematics ought to furnish exact concepts 
 to physics and physics ought to furnish problems for mathe- 
 matics. Thus both subjects would be enlivened and strengthened. 
 
 1 Die Mathematik in den Physikalischen Lehrbiichern, von Dr. H. E. 
 Timerding. O. Professor an der Technischen Hochschule in Braunschweig. 
 Leipzig and Berlin, 1910. 
 
 90 [156
 
 I 57] The Present Teaching of Mathematics in Germany 91 
 
 It is not necessary to weaken mathematics or to lose the ob- 
 jectivity of physics. 
 
 The physics teacher of to-day has a difficult task, because he 
 cannot take for granted that his pupils possess a definite amount 
 of knowledge of mathematics. This is especially true in the 
 university where the amount they do have varies so much that 
 it is hard to adapt the instruction to the needs of all. He cannot 
 cater to the well-prepared but must adapt his instruction to fit 
 the needs of the majority. A reform has already begun in the 
 university, but it has not yet affected the lower classes to any 
 extent. 
 
 The discussion which follows is based solely upon the ex- 
 amination of text-books. The author realizes that good teach- 
 ing is as essential as a good text, but an investigation of the work 
 of teachers would necessarily be very limited. Reports from 
 teachers upon the methods used would probably confirm the 
 statements of the author. This examination has been confined 
 in general to the most popular German texts, the exceptions 
 being in the case of those that are remarkable for any peculiari- 
 ties, regardless of their popularity. 
 
 The problem, then, as the author sees it, is to examine the 
 material at hand to get the scope and character of that part of 
 physics which is usable in mathematics. The mathematics of 
 physics is distinctly geometric, as physical phenomena take place 
 in space. To get the scope and character of the mathematics 
 which has grown from physics, as it is at present, we must first 
 consider the historical development of physics. 
 
 Physical problems came first and the mathematical treatment 
 had to be sought ; but this mathematical development was prac- 
 tically finished when physics texts began to be written. In this 
 development has physics had any influence on modern mathe- 
 matical methods, or has mathematics vitally influenced physics 
 or physics instruction? What was the origin of present mathe- 
 matical methods in physics? Where did they come from, and 
 what was physics before this time? These are some of the ques- 
 tions to be answered. 
 
 The development of the physics texts along mathematical lines 
 begins with the seventeenth century. Physics ceased to be a
 
 92 Teachers College Record [158 
 
 branch of Aristotelian philosophy and became a branch of nat- 
 ural philosophy with Galileo. The physical text-book originated 
 with Descartes, an opponent of Galileo's ideas. One of the 
 earliest texts was published in 1672 in Amsterdam, by Rohault. 
 Both Rohault and Descartes treated the subject quite fantastically. 
 
 The mathematics of those days was only a methodical dis- 
 cipline, and was first used as a dominant method in physics by 
 Newton. Newton's " Mathematical Principles of Natural Phil- 
 osophy " was due, however, in a considerable measure, to Des- 
 cartes. The first result of Newton's influence was a book written 
 in 1721 by Gravesande, " Physices Elementa Mathematica Ex- 
 perimentis Confirmata," published in Leiden. Gravesande in 
 this work states that physics belongs to mathematics. In 1725, 
 at London, Desaguliers published his " Course of Experiments 
 in Philosophy " as a result of Newton's influence and lectures. 
 Desaguliers declared that physics without observation and ex- 
 periment is useless, but that we must bring geometry and arith- 
 metic to our aid unless we wish these observations and 
 experiments to be pure guesswork. To become a physicist one 
 must know mathematics, although Newton's principles can be 
 imparted to the general public without it. This is in accord 
 with the ideas of modern teachers. 
 
 In Segner's " Introduction to Natural Philosophy," published 
 at Gottingen in 1746, stress is laid on geometry, which the author 
 declares to be the foundation of all natural science, although 
 the more difficult and less known principles of the subject may 
 be avoided. Arithmetic may sometimes be used in place of 
 geometry. 
 
 The mathematical ideal pervaded the natural philosophy of 
 the eighteenth century. A certain group of scholars, however, 
 worked along quite different lines from the rest and assumed 
 an attitude of dilettanteism. The work of this group was signifi- 
 cant, as it paralleled the dawn of the modern conception of 
 electricity and magnetism which completely revolutionized the 
 methods of physical instruction. Thus we find that after a long 
 dominion of the mathematical method, the inductive experimental 
 method came into the foreground. The theory of electricity was 
 not taken up in texts until its susceptibility to mathematical treat-
 
 I 59l The Present Teaching of Mathematics in Germany 93 
 
 ment seemed certain; but professional men were inclined to 
 regard it as play rather than an earnest science. 
 
 The first part of the nineteenth century saw many new texts. 
 Gottfried Fischer's " Lehrbuch der Mechanischen Naturlehre," 
 published in 1805, is the basis for many of our modern texts. 
 It was not used to any extent in Germany, but it was translated 
 into French and became a very popular text with the French 
 people. 
 
 It is interesting to consider the origin, growth, and methods 
 of the text-books in physics. A text-book is the outgrowth of 
 a definite need in teaching, and this need is especially great in 
 physics. As the teaching of physics begins in the lower grades 
 and is carried into the university, the number of texts on this 
 subject increases with little improvement in quality. 
 
 Certain characteristics are common to all, conservatism, 
 poor diagrams and pictures, and lack of graphic work. 
 Conservatism arises from the fact that many teachers aim to 
 teach the subject just as it was taught to them. They try to 
 adapt new material to old methods, and prefer a text that clings 
 to the old traditions and resembles as nearly as possible the one 
 they used. The teacher also prefers a text that makes the in- 
 struction as easy as possible. This is especially true in the 
 university, where the lecture method is used and where the aver- 
 age student may not be helped at home. For this reason, a text 
 is apt to have a long lease of life. The Miiller-Pouillet text is 
 a good illustration. It is sixty years old, and is taken from 
 another eighty years old, although it has been revised. Those 
 books that are not original are most successful ; a writer cannot 
 afford to introduce too many novelties. Mach's book is a fine 
 original text, but it has not been a success largely for this reason. 
 
 Another reason for the obstinate holding to the old ideas is 
 that the amount of matter to be treated is so enormous that a 
 text cannot be written from memory, so that the writers must 
 depend on former texts as a source of material and as a model. 
 Many texts have been written in this way without much thought 
 being put upon the work. However, the present tendency seems 
 to indicate a change for the better. 
 
 Another fault common to physics texts is their treatment of
 
 94 Teachers College Record [160 
 
 the problems involving the calculus. The introduction of the 
 conceptions of velocity marks the critical point in the text-book, 
 because it involves the idea of the infinitesimal. Differential 
 calculus is the only sound basis for a conception of velocity, 
 although this concept may be presented without its aid. In 
 fact, this concept may be easily given to the ordinary student 
 of physics. One may conceive of non-uniform velocity, but it 
 cannot be fully comprehended without the idea of limits. In 
 the newer books which, on principle, avoid any such introduc- 
 tion, the laws of gravity are presented in the form of an em- 
 pirical formula obtained by Atwood's machine; Cruger, for 
 example, does this. Mach gives an abridged empirical table 
 which is worked out in detail in his latest book. 
 
 Inaccuracy in drawings and illustrations is another char- 
 acteristic of the texts. These are of great importance mathe- 
 matically, as may be shown by a consideration of the kinds 
 usually shown in texts. The first thing considered by the text- 
 book writer is the expense, and this has led to the use of many 
 poor drawings and many illustrations so old as to have no value 
 to the pupil. In an elementary text by Koope-Husmann there 
 is a picture of a rainbow with large drops of water visible. 
 Gravesande has a figure of a steam engine and of other machines 
 over two hundred years old. A picture of a primitive type of 
 locomotive was in use for over fifty years. Only in the very 
 recent books are there any really new drawings or pictures. 
 
 One method for improving the drawings and pictures is for 
 a publisher to have some very good cuts and use the same in 
 all his books. This is done by a house in Braunschweig and thus 
 excellent cuts are used in cheap books. One objection to this 
 plan is that there is a tendency toward their use for ornamental 
 purposes in which case their significance is lost. Mach and 
 Bremer have drawings by the authors which are very poor. 
 Kleiber-Karsten have many drawings to bring out important 
 facts, but they are of inferior quality. There are advantages in 
 having the author make his own drawings and in having many 
 of them, so this method must not be discarded too readily. La 
 Grange boasted of never having used a figure in his " Elements 
 of Mechanics."
 
 161] The Present Teaching of Mathematics in Germany 95 
 
 For anything that the teacher himself cannot show, the 
 pictures ought to be good; for example, in the study of 
 large machines or cloud formation. " Schematische " figures 
 may be used in place of real figures. This is the case when 
 the mathematical development is brought in. This development 
 should not be carried to a point where it will kill the living con- 
 ception of the thing and make the pupil think in mathematical 
 abstractions. These drawings should be as simple as they can 
 be made. Properly, every author of a text-book should either 
 be a good draughtsman or should collaborate with one. It 
 is of great importance that every drawing shall be true in perspec- 
 tive ; otherwise the pupil will get wrong ideas of geometric con- 
 struction. For example, very few books properly represent the 
 sphere, meridians and parallels of longitude. Warburg has a 
 false perspective drawing of a cone which was taken from a 
 correct drawing in Helmholtz. If the pupil does not have exact 
 pictures, how can he draw correctly, or how can he learn 
 geometry ? 
 
 It is only recently that any importance has been given to the 
 kind of figures and pictures in a text-book. Correct drawings 
 have a great educational value, because the student^ is inspired 
 by the idea that there is a connection between all branches of 
 human knowledge, and correct drawings enable him to get some 
 idea of this connection. For example, accurate drawings of 
 rays of light going into water, of waves of sound, and of the 
 refraction of rays of light in raindrops, give him ideas of 
 geometric curves. 
 
 Graphic representations are of an importance only recently 
 recognized. Without exaggeration, all important functional phe- 
 nomena that occur in nature may be pictured by curves. The 
 graphic representation of different functions is one of our most 
 important means of teaching mathematics, therefore physics 
 aids greatly in mathematical instruction since various physical 
 phenomena lend themselves to graphic treatment. England is 
 largely responsible for the introduction of the use of graphic aid% 
 in instruction in physics. In Germany to-day, physics and graphic 
 representations are so far apart that there is no possibility of 
 applying the graph to physics, although it was used in a treatise
 
 96 Teachers College Record [162 
 
 on natural philosophy nearly fifty years ago. When we find 
 any use of the graph it is in connection with thermodynamics. 
 The isothermic lines are generally used to give an idea of tem- 
 perature and of the critical point. Clausius was the first man in 
 Germany to do this kind of work and he has been very much 
 interested in it. 
 
 The paper just summarized is interesting for one thing, 
 namely, we find the Germans so far behind us in the matter of 
 texts. For some time the necessity for good drawings and for 
 good illustrations in texts has been recognized in this country, 
 and to-day we have many good texts which are very satisfactory 
 in this respect. 
 
 We have in this country two bodies of physicists those inter- 
 ested in correlating mathematics and physics, and those interested 
 in keeping them as far apart as possible ; but the college exam- 
 inations settle quite definitely the type of work that must be 
 done by schools that attempt to prepare for college, so most 
 of our physics is the so-called mathematical physics. We will 
 agree, I think, that a sufficient amount of mathematics should 
 precede physics, but as to how much that should be we are 
 uncertain, as are the Germans. The ideas of the calculus may 
 be taught much earlier than we teach them, and to-day ele- 
 mentary calculus is being given in some high schools. In gen- 
 eral, Germany seems to be doing as little as we are to correlate 
 closely the mathematics and the physics of the schools.
 
 CHAPTER XIV 
 
 GOVERNMENT EXAMINATIONS IN PRUSSIA AND 
 THE NORTH GERMAN STATES 
 
 Cilda Langntt Smith and Katherine Simpson 
 
 The report here reviewed 1 gives the important regulations that 
 were passed in regard to the state examinations in Prussia and 
 the United North German States from 1810 to 1898, and also 
 the regulations concerning the jexaminations in Braunschweig and 
 Mecklenberg. Dr. Lorey shows that the principal reason why 
 Germany has accomplished so much in the mathematical field 
 is because her educators early conceived the idea of gradually 
 increasing the mathematics requirements for the teachers, and 
 the result of this policy is that the schools of to-day have thor- 
 oughly equipped teachers who have a broader knowledge of the 
 subject than the bare contents of the curriculum. 
 
 The higher schools of Prussia and the United North German 
 States were established many years before 1810. These higher 
 schools were generally founded by the church and certain com- 
 missions. Their teachers were the clergymen who, while waiting 
 for better paid positions, regarded this as the most profitable 
 way to employ their time. The order of the clergy was pre- 
 ferred because the teaching profession at that time was con- 
 sidered one of comparatively low rank. 
 
 Two minor laws in regard to the requirements of the teachers 
 were passed before 1810 ; the one in 1718, which decreed that all 
 the teachers in the German and Latin schools must pass an 
 examination before a committee, the other in 1787, which decreed 
 that every teacher before being admitted to the teaching pro- 
 fession must hold a teacher's certificate. But the edict of 1810 
 was the first one which was really enforced. The purpose of 
 
 1 Staatspriifung und Praktische Ausbildung der Mathematiker an den 
 Hoheren Schulen in Preussen und Einige Norddeutschen Staaten, von 
 Dr. Wilhelm Lorey, Pro-rektor der Kge. Oberrealschule in Minden, 1911. 
 
 163] 97
 
 98 Teachers College Record [164 
 
 this edict was to prevent incapable teachers from entering the 
 teaching profession. It must be understood that the states in 
 which this edict was to take effect were divided into three de- 
 partments of instruction, i.e., the departments of Berlin, Breslau, 
 and Konigsburg. The examinations were held before commit- 
 tees chosen by these departments. The following teachers 
 were required to take the examinations: future teachers of 
 the public schools who were preparing pupils for the second 
 and third classes of the Gymnasium and Realschule (which 
 classes are the same as our sixth and seventh grades respec- 
 tively) ; teachers who were planning to teach in private schools; 
 and teachers of the public schools who were to prepare the pupils 
 for entrance to universities. Those who were not required to 
 take the examinations were teachers in the elementary schools 
 (the Folks-schulen and Biirger-schulen) and young graduates 
 of the universities, who were planning to teach only for a 
 short time. One can scarcely judge the standard of the uni- 
 versities at this time, because the requirements, as outlined in 
 this edict, were very indefinite. The candidate was required to 
 have not only a general education but also a more thorough 
 knowledge of history, mathematics, and philosophy. These re- 
 quirements did not go into effect, however, until 1813, as it was 
 considered that it would take the universities three years to 
 prepare the candidate for meeting these requirements. 
 
 It was not until the ordinance of 1831 that a very definite 
 statement of the requirements was made. These regulations 
 gave the candidate the privilege of taking one of four different 
 examinations, according to the kind of position he desired. 
 
 1 i ) The " pro f acultate docendi " examination, which was the 
 most definitely outlined examination and was considered the most 
 important. 
 
 (2) The " pro loco " examination, which was not found prac- 
 tical and was not considered legal after 1866. 
 
 (3) The "pro ascensione " examination, which also was not 
 open to candidates after 1866. 
 
 (4) The " colloquia pro rectoratu " examination, which even 
 to-day is one of the examinations for which a candidate may 
 prepare.
 
 165] The Present Teaching of Mathematics in Germany 99 
 
 In order that a candidate might take the " pro facultate 
 docendi " examination, it was necessary that he have knowledge 
 of all subjects, i.e., languages (Greek, German, Latin, French, 
 and Hebrew), mathematics, physics, German history, geography, 
 mythology, literature of the Greeks and Romans, philosophy, 
 pedagogy. 
 
 Having passed an unconditioned " facultas docendi," the can- 
 didate had the privilege of teaching in all classes of the Gym- 
 nasium, provided he also showed a special knowledge of at least 
 two of the ancient languages, his own language, mathematics, 
 nature study (botany, mineralogy, chemistry, and zoology), his- 
 tory, and geography. Wishing to teach mathematics in the Gym- 
 nasium, a candidate, in order to meet the requirements of the 
 " facultas docendi," must have a thorough knowledge of the 
 following subjects: 
 
 1 i ) Elementary geometry and common arithmetic for instruct- 
 ing in the lower classes (which are the same as our fourth, fifth, 
 and sixth grades). 
 
 (2) Geometry, including both plane and solid, plane trigonom- 
 etry, and common arithmetic for teaching in the middle classes 
 (which correspond to our seventh and eighth grades, and the 
 first year in the high school). 
 
 (3) Higher geometry, analysis of infinity, applications of 
 mathematics to astronomy and physics (physics and mathematics 
 being correlated) for teaching in the higher classes (which cor- 
 respond to the last three years of our high school course). 
 
 For the first time, one finds the study of the science of edu- 
 cation emphasized. Psychology, logic, and philosophy, including 
 its history and changes since the time of Kant, formed an inti- 
 mate part of the candidates' studies. Until the edict of 1831 
 mathematics and nature study had been correlated, but it was 
 then decided to separate the two subjects, as it was thought 
 impossible to find teachers who were thoroughly efficient in 
 both departments. Physics was considered more as an experi- 
 mental study than a mathematical study. The candidate was re- 
 quired to have a general knowledge of the entire subject, to 
 know its phenomena in nature and its laws, to be familiar with 
 the apparatus for the study of physics, and to be able to perform
 
 ioo Teachers College Record [166 
 
 the ordinary experiments in class. It was in this edict that 
 drawing was first emphasized. 
 
 The second and third examinations, i.e., "pro loco" and "pro 
 ascensione," were of minor importance. The aim of the " pro 
 loco " examination was to show the ability of the candidate for 
 certain positions in special schools. The "pro ascensione" exam- 
 ination was for the teacher who wished to teach in the higher 
 schools. 
 
 The fourth examination, i.e., " colloquia pro rectoratu " was to 
 determine whether a candidate for the rectorship in the higher 
 schools was capable of filling the position. In this the candidate 
 was examined partly in Latin and partly in German upon 
 pedagogical and didactic subjects. One is no longer in doubt as 
 to one of the reasons why Germany has been so successful in 
 building up a school system of such great force, when he learns 
 of the many requirements of the rector of the higher schools 
 even as early as 1831. 
 
 Each candidate who took one of the regular examinations was 
 required to have six months of observation either in a Gym- 
 nasium or Realschule. The Oberlehrers (teachers in the higher 
 classes) especially were very much dissatisfied with the recent 
 ordinance. They were greatly agitated over the question of 
 being allowed to enter a pedagogical seminary, such a school as 
 one finds in Teachers College, Columbia University, and have 
 the work done there count towards a degree. 
 
 Regardless of the fact that mathematics was gradually develop- 
 ing in the universities, its progress in the Gymnasien was much 
 slower. It is a lamentable fact that Gauss of the University of 
 Gottingen, at the time of the passing of the ordinance of 1866, 
 though doing so much to promote pure mathematics at the Uni- 
 versity, took no pains to develop in his students a knowledge or 
 interest in the mathematics of the secondary schools. His atti- 
 tude towards the advancement of mathematics in the secondary 
 schools was the attitude of most of the university professors of 
 mathematics at that time. Indeed, the first examiner for mathe- 
 matics was not Gauss, but Thibaut, the professor of philosophy 
 at the University of Gottingen. The advancement of the schools 
 was entirely in the hands of the classics instructors who tried to
 
 167] The Present Teaching of Mathematics in Germany 101 
 
 retard mathematics as much as possible. It was almost unheard 
 of at that time for a mathematician to be the rector of a higher 
 school. 
 
 The ordinance of 1866 was one for which the teachers had 
 long looked and hoped. The requirements for the different 
 classes were as follows : 
 
 I. LOWEST CLASSES: plane geometry, stereometry, common 
 arithmetic, algebra, and method of instruction in arithmetic. 
 
 II. MIDDLE CLASSES: plane and solid geometry, plane and 
 spherical trigonometry, algebra up to equations of the third and 
 fourth degrees, analytic theory of straight lines and planes as 
 applied to conies, fundamental operations of the differential and 
 integral calculus, and the main laws of statics. 
 
 III. HIGHER CLASSES: research into higher geometry, higher 
 analysis, and analytic mechanics. This work was to be the basis 
 for independent research. The candidate in mathematics was 
 required to be examined in nature study (chemistry, zoology, 
 mineralogy, and botany). The candidate in nature study had 
 to meet the requirements in the mathematics of the middle classes. 
 The main point emphasized in this edict was that the candidate 
 must have a thorough knowledge of the technical language of 
 the specialized subject. 
 
 This ordinance of 1866 regulated the granting of three grades 
 of certificates, for which the requirements were as follows: for 
 the first grade certificate, mathematics and physics for the highest 
 classes, and nature study for the middle classes ; for the second 
 grade certificate, a capacity for mathematics, physics, and one 
 of the nature studies for the middle classes ; for the third grade 
 certificate nothing but a complete knowledge of the specialized 
 subject. The teachers were very much opposed to these certifi- 
 cates. They considered it detrimental to the specialized subject 
 to be required to have such a general knowledge in order to 
 receive a first grade certificate. 
 
 Another ordinance was issued in 1877, but the mathematics 
 requirements remained practically the same with the exception 
 that a candidate for the higher classes was required to know more 
 applied mathematics. The fact was emphasized that the candi- 
 date should have a clearer insight and a more scientific knowl-
 
 1O2 Teachers College Record [168 
 
 edge than was necessary for the teaching of the subject. It was 
 after this ordinance was issued that the university professors 
 became much interested in the development of mathematics in 
 the higher schools. One of the big questions that was being 
 agitated at that time, as it is to-day, was whether the stress 
 should be laid upon the method of instruction or upon the 
 scientific knowledge. 
 
 One of the prominent mathematicians of the day said that if 
 the student were allowed to devote all his time to acquiring a 
 knowledge of his science instead of being required to give so 
 much of it to methods of the recitation, he would have a firm 
 foundation upon which to build, and very soon would acquire 
 the method by means of experience. 
 
 During the ten years following the regulations of 1887, Pro- 
 fessor Klein was the strongest force directing the tendencies 
 of mathematics teaching in Prussia. This he accomplished 
 through his special vacation courses for teachers which were 
 offered in the universities. Lectures on some phase of mathe- 
 matics teaching, especially on applied mathematics, were deliv- 
 ered by him to the students taking these courses, and to the 
 university professors. The University of Gottingen seems to 
 have been the centre of his influence, establishing courses in 
 science and mathematics, and encouraging the teaching of 
 applied physics. It was Professor Klein's enthusiasm for 
 reform that brought him, during this period, to the United States 
 to attend a congress of mathematicians at the World's Fair 
 in Chicago. His lectures on " The Present State of Mathematics " 
 and his plea for an international union of mathematicians con- 
 tained the ideas which he brought as his message to the congress. 
 
 By 1898 the tendencies in mathematics teaching had taken 
 definite form. That year the next regulations in regard to teach- 
 ers' examinations were passed by the government of Prussia. 
 The most important of the new requirements were: first, that 
 teachers of pure mathematics in the first five of the lower 
 grades should have a thorough knowledge of primary mathe- 
 matics, plane geometry, analytic geometry, integral and differ- 
 ential calculus ; second, that teachers of higher mathematics must 
 be masters of higher geometry, arithmetic (theory of numbers),
 
 169] The Present Teaching of Mathematics in Germany 103 
 
 algebra, higher analysis and analytic mechanics in order to be 
 qualified in pure mathematics; third, that they must know 
 applied mathematics. This included descriptive geometry as 
 far as the rules of central projection, mathematical methods for 
 technical mechanics especially graphic statics, and higher and 
 lower geodesy. Another regulation was that mathematics must 
 always be connected with physics; and it was also decided that 
 knowledge of astronomy was to be expected of a candidate, 
 though no formal examination should be required. One con- 
 cession was made; namely, that the candidate found deficient in 
 the examination could be permitted to teach on condition that 
 he prove himself qualified to teach one of the higher subjects 
 or two of the lower ones. 
 
 These regulations of 1898 are of special interest because they 
 are still in force, no radical changes having been made by the 
 government. However, this is not to be construed as a sign that 
 such a standard has been reached that no further effort toward 
 improvement is considered necessary. The facts show that the 
 various provinces have been kept busy with problems demanding 
 attention. The commissioners who make up the examinations 
 have met yearly since 1900 to discuss changes in the course of 
 study. 
 
 In 1900 at a convention of university professors called for the 
 purpose of discussing some measures of reform for the higher 
 schools of Berlin, one question proposed was, What can be done 
 to raise the standard of technical and applied mathematics? At 
 the same conference the growing interest in the teaching of mathe- 
 matics was attested by the presence there of a greater number of 
 mathematics teachers than had been present at any previous simi- 
 lar gathering. It is significant too that the subject of Professor 
 Klein's lecture to the teachers during the vacation course that 
 year was technical and applied mathematics. Again, in 1902, 
 he was emphasizing two points especially: first, that candidates 
 should be required to make good drawings with explanations 
 of solutions ; second, that models and construction work were 
 necessary to teach technical mathematics properly. Many were 
 interested in the teaching of applied mathematics. There was 
 a feeling on the part of some against separating it from the pure 
 mathematics lest the candidate shun the difficulties of the pure
 
 IO4 Teachers College Record [170 
 
 mathematics in favor of the applied mathematics. Others, of 
 whom Studys is representative, think that the applied mathe- 
 matics is of value only when connected with the pure mathe- 
 matics. 
 
 A second question is one which has provoked discussion in 
 our country as well as in Prussia: What shall be the minimum 
 of general culture required of a candidate specializing in mathe- 
 matics? Studys insists that no subject except mathematics should 
 have any claim. He dismisses with scorn the idea of rejecting 
 a candidate who has passed his examinations in mathematics, 
 merely because he has failed in religion ! 
 
 How have the new tendencies and higher standards affected 
 the number of candidates in mathematics? We find that the 
 number has varied directly with the difficulties of the require- 
 ments. In 1909 there were 280 candidates, to 25 in 1839. Dur- 
 ing the years from 1907 to 1909 the increase has been 66^3 per 
 cent. Assuming that the ratio of supply and demand has not 
 varied unreasonably, these statistics show a marvelous growth 
 in mathematics teaching in Prussia. 
 
 The need of boards of examiners arose with the regulations 
 of 1816 requiring teachers to pass certain examinations. There 
 are five of these boards in all, with centres of organization dis- 
 tributed among the provinces. Each board of examiners is 
 subject to the over-president of the province in which it is 
 located. The members are appointed yearly by the Minister of 
 Instruction of Prussia. For some time the examiners were 
 without exception university men, but gradually public school 
 teachers were made eligible. However, the wisdom of this step 
 has been questioned on the ground that the public school men 
 have in the majority of cases proved incompetent. It is claimed 
 by others that examinations under university professors have 
 often been unfair, that the public school men are better fitted 
 to conduct the examinations since they understand better the 
 requirements of the schools and are less bound by formulas. 
 The latter opinion has evidently prevailed. In some of the 
 provinces the chairman of the board must be a public school 
 teacher. The Minister of Instruction has spoken in favor of 
 retaining them ; and the educational commission of 1907 held 
 that examiners should all be public school men.
 
 171 ] The Present Teaching of Mathematics in Germany 105 
 
 The message for us in this report is written large. We see 
 that time and energy, and above all, the thought of some of the 
 best minds of Germany have entered as factors in determining 
 these standards for the examinations of the public school teach- 
 ers. That the superior excellence of their schools has been a 
 result of all this care, we cannot doubt. If we are to keep abreast 
 of the present movements in education we must introduce into 
 our schools more of the mathematics demanded by the practical 
 problems of modern life. But we cannot introduce into our 
 schools such subjects as descriptive geometry and differential 
 calculus until we have teachers for them. The need of higher 
 standards for our schools is too well known to require dis- 
 cussion. The question with us is, How can these reforms be 
 accomplished? We cannot look to a centralized government to 
 do this, nor is it probable that this method, if possible, would 
 be for the best interests of education in our country. The start- 
 ing point for this reform is thought to be in our larger cities 
 since they can exert such a wide influence, have so much free- 
 dom in the government of their schools, and are directly re- 
 sponsible for the welfare of so large a part of our population. 
 A few of them have already started the ball rolling. New 
 York, for instance, now requires for teachers of mathematics 
 an examination in trigonometry, analytic geometry, and the 
 calculus, beyond the subjects included in the course of study. 
 With a view to encourage this spirit of progress, and to arouse 
 a wholesome rivalry among the citifcs, Commissioner Claxton 
 is preparing for publication statistics from reports as to what 
 each of the cities is doing. 
 
 But whatever machinery may be put in motion to produce these 
 reforms in mathematics teaching, the power that turns the wheels 
 will be applied by institutions typified by Teachers College. For 
 from these institutions should go out teachers who are informed 
 not only as to how mathematics ought to be taught and what 
 mathematics ought to be taught, but, most important of all, who 
 themselves know mathematics. Therefore we consider the great- 
 est good which it is within the power of these institutions to 
 accomplish is by wise instruction to create an enthusiasm for a 
 subject, which by its impetus alone will carry a student far 
 beyond the limits of bare " requirements " for teaching.
 
 CHAPTER XV 
 DESCRIPTIVE GEOMETRY IN THE REALSCHULEN 1 
 
 Louise Eugenie Harvey and Jessie Mae Reynolds 
 
 The material for this report was collected by Dr. Ziihlke from 
 three sources, the official regulations on the subject of line- 
 drawing and descriptive geometry, the literature in the depart- 
 ment of mathematics, and the results of a tour of investigation 
 undertaken during the previous year. This tour of investigation 
 carried Dr. Zuhlke to about thirty German schools. In order to 
 compare conditions prevailing among them with those which had 
 a marked influence on the course of study in southern Germany, 
 he also visited four Austrian Oberrealschulen. 
 
 The stress laid upon the subject of descriptive geometry in 
 the German schools and the claims made for it are worthy of 
 consideration by us who have hardly ever thought of the subject 
 as one of general importance. The report discusses the place 
 of line-drawing and descriptive geometry in the mathematics 
 curriculum of the Realschulen, giving details concerning the 
 number of hours devoted to them in the various institutions and 
 sketching some of the plans of study. Under the heading of 
 " procedure in teaching," sundry questions of method are treated. 
 The use of models and the conditions in the rooms assigned to 
 drawing are briefly discussed. Finally the training of the 
 teacher is considered. 
 
 THE PLACE OF LINE-DRAWING IN THE CURRICULUM 
 The exact designation of the subject does not seem to be 
 quite settled in Germany; that is, there is some question as to 
 the department of mathematics to which it belongs. Line- 
 drawing and descriptive geometry are sometimes called concrete 
 
 1 Der Unterricht im Linearzeichnen und in der Darstellenden Geometric 
 an den deutschen Realanstalten, von Dr. Paul Zuhlke, Leipzig und Berlin, 
 IQII. 
 
 106 [172
 
 173] The Present Teaching of Mathematics in Germany 107 
 
 mathematics, sometimes practical mathematics, and sometimes 
 pure mathematics. Some wish to make descriptive geometry 
 part of the course in line-drawing. The opinion of the best 
 educators seems to be that skill in drawing is essential in 
 descriptive geometry, and that if the correct result is to be 
 produced, both sides of the subject, the theory and the tech- 
 nique, should be equally considered. Then comes the question 
 of precedence: Should skill in drawing be required first, the 
 theoretical work following, or should theory be introduced at 
 the beginning? And would this latter method spoil much of 
 the delight in the beauty of the work? The former view, that 
 of requiring skill in drawing first, is the more general one. 
 
 The exact relation of line-drawing and descriptive geometry 
 to the remaining branches of study has been determined in vari- 
 ous parts of Germany. In the Realschulen of Prussia, line- 
 drawing was a separate subject from 1901 to 1909, although 
 it was cited under the heading of " drawing " in the official pro- 
 gram of study. In 1908 it was divided into two sections, mathe- 
 matics and drawing, each class having two hours of instruction 
 a week from a drawing teacher and the other three hours from 
 a mathematics teacher. It is in Wiirttemberg, Saxony, and Baden 
 that line-drawing and descriptive geometry are most closely con- 
 nected with mathematics, and, with the exception of five hours' 
 work in Wiirttemberg, it is everywhere obligatory. In the Ober- 
 realschulen of Hamburg descriptive geometry is elective in the 
 following sense: four hours per week of instruction have been 
 prescribed, two of which are designated as freehand drawing and 
 two as descriptive geometry. Of these four hours, the student 
 must elect two, but it is left to him whether he will take one 
 from each group, or both from one. Dr. Ziihlke comments on 
 this arrangement with the caustic remark, " In any case it 
 can be imagined that a graduate of a Hamburg Oberrealschule 
 knows nothing of descriptive geometry." 
 
 The division of what is called line-drawing, and especially 
 its distribution among different teachers, is thought unfortunate. 
 The administration seems to sanction the arrangement, however, 
 judging from a decree in which it is stated that so long as cer- 
 tain difficulties continue, things must remain as they are.
 
 io8 
 
 Teachers College Record 
 
 [174 
 
 MATHEMATICS CURRICULUM IN THE REAL-SCHOOLS 
 The number of hours devoted to line-drawing and descriptive 
 geometry will perhaps be best understood from the following 
 table : x 
 
 CLASS IN GKRMAJ 
 CORRESPONDING 
 TO ABOUT: 
 
 w SCHOOL 
 
 IV 
 
 6th 
 school 
 year 
 
 UIII 
 7th 
 school 
 year 
 
 O III 
 
 8th 
 school 
 year 
 
 U II 
 
 9th 
 school 
 year 
 1st 
 H. S. 
 
 O II 
 
 10th 
 school 
 year 
 2nd 
 H. S. 
 
 UI 
 
 llth 
 school 
 year 
 3rd 
 H. S. 
 
 I 
 
 12th 
 school 
 year 
 4th 
 H. S. 
 
 IN OUR SCHOOLS 
 
 Prussia 
 Realschule Line-drawing 
 Oberrealschule / Line-drawing 
 and Realgymn. { Descriptive geom. 
 
 - 
 
 2 
 
 2 
 2 
 
 *toto 
 
 2 
 
 2 
 
 * 
 
 2 
 
 * 
 
 Bavaria 
 Oberrealschule 
 Realgymnasium 
 
 Line-drawing 
 Technical drawing 
 Descriptive geom. 
 Line-drawing 
 Descriptive sreom. 
 
 1 
 
 2 
 
 * 
 2 
 
 * 
 2 
 
 * 
 
 * 
 
 1 
 
 _ 
 
 * 
 
 
 
 * 
 2 
 
 2 
 
 Wurtltmberg 
 Oberrealschule ( Geometric drawing 
 and Realgymn. ) Descriptive geom. 
 since 1910 [ Projective drawing 
 
 - 
 
 * 
 
 * 
 
 1 
 
 * 
 
 * 
 
 * 
 2 
 
 * 
 2 
 
 Saxony 
 Oberrealschule f 
 Realgymnasium j Line-drawing 
 Realg. Abt. of] (descriptive geom.) 
 Reformgymn. [ 
 
 - 
 
 - 
 
 - 
 
 1 
 
 2 
 2 
 
 2 
 2 
 3 
 
 2 
 2 
 3 
 
 Baden 
 Oberrealschule 
 (now Reahzym. 
 in Mannheim) 
 Realgymnasium 
 (Reform plan) 
 
 Instruction 
 in 
 representation 
 
 - 
 
 - 
 
 - 
 
 2 
 
 2 
 2 
 
 2 
 2 
 
 2 
 2 
 
 Hesse 
 Oberrealschule / Geometric 
 Realgymnasium \ drawing 
 
 - 
 
 - 
 
 - 
 
 1 
 
 1 
 
 2 
 
 2 
 
 Hamburg 
 Oberrealschule 
 
 Descriptive geom. 
 
 - 
 
 - 
 
 - 
 
 - 
 
 2 
 
 2 
 
 2 
 
 AUace' 
 Oberrealschule . 
 
 
 - 
 
 * 
 
 * 
 
 * 
 
 2 
 
 2 
 
 2 
 
 
 
 The most liberal obligatory courses are those of the Ober- 
 realschulen of Baden, which devote to the subject two hours per 
 week during the last four years. The direct opposite to these 
 are the Realgymnasien of Hesse which evidently do not consider 
 the subject at all. The course as an optional is most extensively 
 pursued in Prussia, where two hours per week during the last 
 
 1 In this table an asterisk means that the official program calls for 
 obligatory instruction in the respective classes, but that no fixed hours 
 have been designated.
 
 J 75] The Present Teaching of Mathematics in Germany 109 
 
 five years are given to it. Up to 1904, descriptive geometry was 
 obligatory in the seventh, eighth, and ninth classes of the Ober- 
 realschulen of Wiirttemberg, where two hours per week in the 
 seventh year, four hours in the eighth, and four hours in the 
 ninth year, were required. In January, 1904, the course was 
 extended one hour in the highest class. Through an arrange- 
 ment made in April, 1910, a part of this " reform " was discon- 
 tinued, in that it was decided that in the obligatory course of 
 the eighth and ninth classes, descriptive geometry should be 
 treated in close connection with analytic geometry, to which two 
 hours per week instead of three were now given. 
 
 Dr. Ziihlke states that a comparison of the number of hours 
 with those of the Austrian Oberrealschulen leaves much to be 
 desired, for in Austria one finds a total of fifteen hours per 
 week of obligatory work in geometric drawing and descriptive 
 geometry. Indeed the study of solids in space is given much 
 more time in Austria than in Prussia. The proportion of work 
 done in both states may be seen from the following table: 1 
 
 AUSTRIAN 
 
 
 
 
 
 
 
 
 
 OBERREAI>SCHULEN 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 VII 
 
 Total 
 
 Calculation and Math- 
 
 
 
 
 
 
 
 
 
 matics 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 ISem. 4 
 
 5 
 
 26 
 
 Geometric drawing and 
 descriptive geometry 
 
 
 2 
 
 2 
 
 3 
 
 3 
 
 II Sem. 3 
 3 
 
 2 
 
 (25) 
 15 
 
 Freehand drawing. . . . 
 
 4 
 
 4 
 
 4 
 
 3 
 
 3 
 
 2 
 
 3 
 
 23 
 
 PRUSSIAN 
 OBERREALSCHULEX 
 
 VI 
 
 V 
 
 IV 
 
 UIII 
 
 OIII 
 
 UII 
 
 Oil 
 
 UI 
 
 01 
 
 Total 
 
 Calculations and Mathe- 
 matics 
 
 5 
 
 5 
 2 
 
 6 
 2 
 
 6 
 2 
 
 5 
 2 
 
 5 
 2 
 
 5 
 2 
 
 5 
 
 2 
 
 5 
 2 
 
 47 
 16 
 
 Freehand drawing 
 
 
 1 In Austria the Oberrealschulen are seven-class schools. The students 
 of the lowest class must be ten years old, about as old as the students 
 of the fifth class in the regular German Realschulen. The students of 
 the highest (7th) class are, as a rule, older than those of the ninth class 
 in the German schools, as a result of the " Classification Examinations " 
 in Austria.
 
 no Teachers College Record [176 
 
 The southern states stand somewhere between Prussia and 
 Austria with respect to the time devoted to the subject. For 
 example, Bavaria, in the nine years, devotes to freehand draw- 
 ing and line-drawing a total of twenty-six hours of obligatory 
 work, as compared with a total of twenty-three in seven years 
 in Austria for line-drawing alone. 
 
 Besides the statement of the average time devoted to line- 
 drawing and descriptive geometry, some of the official programs 
 of study are given. That of 1910 for the Prussian Oberreal- 
 schulen and Gymnasien requires the following courses : 
 
 UII : Perspective drawing of solid figures. 
 
 UI: Elements of descriptive geometry. 
 
 On the other hand the following courses in line-drawing are 
 elective : 
 
 OIII : Practice in the use of the compasses, ruler and drawing- 
 pen, through the drawing of surface designs and other geometric 
 forms. 
 
 UII : Geometric description of single solids in different views, 
 with sections and the development of surfaces. 
 
 Oil and I: Further study in descriptive geometry; study of 
 shading and perspective. 
 
 WORK IN THE HUMANISTIC GYMNASIEN 
 
 The report contains, in an appendix, a brief statement of the 
 work in space-perception done in the humanistic Gymnasien. The 
 introduction of the teaching of projection in the course of study 
 is a result of the reform movement. In Prussia the study plan 
 of 1901 made a great advance in this department, in so far as 
 for the first time instruction in the perspective drawing of 
 solids was admitted. Many educators energetically favor having 
 projective geometry in the course of study in the Gymnasien. 
 
 Although a regulated treatment of stereometric constructions 
 can not always be accomplished in the instruction in mathe- 
 matics, still many a good opening for that kind of work remains. 
 For example, in the official program of study for the UII of the 
 Gymnasium, under the work in natural science, is the topic: 
 " Discussion of some of the most important minerals " ; and, 
 under geography, the topic, " Elementary mathematical geogra-
 
 177] The Present Teaching of Mathematics in Germany in 
 
 phy." Both of these departments are in the hands of the teacher 
 of mathematics in the UII of the Gymnasium. So here is an op- 
 portunity for a very easy introductory treatment of parallel pro- 
 jection inserted in the instruction in mineralogy and geography. 
 This work is an actual fact in the Kaiser Friedrich Gymnasium 
 at Frankfurt. The aim of the work, they say, is to introduce 
 all students, and not merely a few, to geometric and especially 
 to stereometric drawing. The technical course in drawing in. 
 V and VI gives practice in using ruler and compasses. 
 
 UIII: Exercises in perspective. 
 
 OIII: Drawing of simple crystals, cubes, octahedrons, etc. 
 
 UII: Analysis of minerals, of square and hexagonal systems 
 of crystallization. Extension of exercises in the drawing of 
 crystals. 
 
 UI: Scientific introduction of projection in solid geometry. 
 
 OI: Cross-section of solids. 
 
 The teacher of mathematics should be given latitude in doing 
 the correlation in this work. 
 
 The study in projection may be discontinued at any time, for 
 even a little of it gives many advantages, among them neatness 
 and elegance in drawing, and training of the eye and hand. Dr. 
 Ziihlke says that the groundwork of instruction in projection 
 and the elements of stereometric drawing must become obligatory 
 in all Gymnasien if they are not to be blamed for turning out 
 
 unpractical youths. 
 
 f 
 
 PROCEDURE IN TEACHING 
 
 Dr. Ziihlke believes that instruction in line-drawing and 
 descriptive geometry should proceed, as in every study, from the 
 near to the remote. Appeal should therefore be made first to 
 the sense-perception. There are, also, certain fundamentals 
 which need no demonstration ; certain intuitions upon which the 
 teacher can easily build. Thus the student should be guided 
 slowly but constantly from intuitive knowledge to acquired 
 knowledge, and, to this end, " not to draw mechanically, but 
 to draw intelligently, must be the highest rule." 
 
 A teacher who begins with too abstract inquiries tires the 
 student ; and weariness is, according to Herbart, the heaviest
 
 H2 Teachers College Record [178 
 
 wrong of any course, " for a man does not live on what he eats, 
 but on what he digests." However, in the upper grades, the 
 desire for abstract thought develops, and here, knowledge must 
 be systematized. Yet Dr. Ziihlke warns the teacher against an- 
 ticipating in any way the academic course of study, for nothing 
 is more pernicious for a young man who has taken descriptive 
 geometry in the high school than the consciousness, when he 
 reaches college work, that he has already had it all. 
 
 In considering further the correct relation between abstract and 
 concrete knowledge, shadow-picturing was given as an example 
 of a much more interesting and instructive lesson than any 
 arrangement of abstract figures. A sound pedagogue, it is said, 
 moves continuously here and there between theory and practice, 
 making a sane use of both. 
 
 Descriptive geometry should not be an isolated subject, for as 
 such it is unfruitful, but should be the connecting link between 
 pure and applied mathematics. In beginning the subject, in- 
 struction should proceed along the lines of the question-develop- 
 ing method. The author warns against too much explanation 
 and talking on the part of the teacher. Papers are put into the 
 hands of the students on which neither the method of solution, 
 nor the result is given, but merely the hypothesis ; or a complete 
 drawing, from which he is to recognize the conditions of the 
 problem, is given him for brief inspection. 
 
 The following is a reproduction of an actual recitation in the 
 UII of an Oberrealschule (which corresponds in time to our 
 first year in high school). In this the students had become 
 accustomed to space conceptions, and had learned to see the 
 drawing completed before it was actually finished. 
 
 TEACHER: What proposition did we consider in the last 
 recitation ? 
 
 STUDENT: We considered the intersection of a straight line 
 and a plane. 
 
 TEACHER: By what means were the two determined? 
 
 STUDENT: The straight line was given by its projections; 
 the plane by its traces. 
 
 TEACHER: Explain the method of solution.
 
 179] The Present Teaching of Mathematics in Germany 113 
 
 (One student draws at the board; several, one after another, 
 explain the solution.) 
 
 TEACHER: We will take up the same proposition to-day, but 
 considered from other data, since in practice a plane is very 
 seldom given by its traces. By what other means can a plane 
 be determined? 
 
 STUDENT: A plane can be determined by three points not 
 
 lying in a straight line, or by a straight line and a point without, 
 or by two intersecting straight lines. 
 
 TEACHER: We wish three points for our plane. Take, on 
 the board, A == (2, i, 3), B = (4, 5, 6), C = (6, 2, 4). 
 
 (A student does this. The board is covered with fine gray 
 lines, which cannot be distinguished at more than two or three 
 meters distance. The student letters, without special request, 
 the projections of the separate points A', A", B', B", C, C".) 
 
 TEACHER: Draw also the projections of the triangle deter- 
 mined by the points A, B, C. (This is done.) Let us consider
 
 U4 Teachers College Record [180 
 
 for a moment the triangle ABC cut out of this copy-book 
 cover, which is blue on the outside and white inside. Then 
 would like or different colors be apparent in the horizontal and 
 vertical projections? 
 
 STUDENT : Different colors would be seen. 
 
 TEACHER: Why do you conclude that? 
 
 STUDENT: The triangle A' B' C' has an opposite sense of 
 revolution to the triangle A" B" C". 
 
 TEACHER : Hold this paper triangle about as the figure would 
 indicate. (A student does this.) 
 
 TEACHER: Now we still need a straight line g which we will 
 determine by two of its points. Take PEEEF (2, 5, 2), Q =E (7, 
 i, 7), and the projection of the straight line determined by them. 
 (This is done; the points P', P" , Q', Q" are marked but lightly, 
 the addition of the letters not being made because of the result- 
 ing indistinctness of the figure. Only g', g" are drawn.) 
 
 TEACHER : Now hold this pointer in the position the straight 
 line would take according to our sketch. (This is done.) Now 
 draw in all the data in your note-books. (The students do this.) 
 
 TEACHER: How will we now proceed? 
 
 STUDENT: We can, as in the last recitation, pass through the 
 straight line g an auxiliary plane at right angles to one of the 
 planes of projection and determine the intersection of this auxil- 
 iary plane with the plane of the triangle ABC. 
 
 TEACHER: What do we call such a plane which is at right 
 angles to one of the planes of projection? 
 
 STUDENT: Such a plane is called the projecting plane. 
 
 TEACHER: Should we choose, in this case, a vertical project- 
 ing-plane, or a horizontal projecting-plane? 
 
 STUDENT: That will depend upon which relation gives the 
 more favorable intersection. 
 
 TEACHER: Now determine which will be the best plan here. 
 (A student steps forward, holds with one hand the paper triangle 
 in the correct position, designates the straight line g by a knitting- 
 needle taken from the teacher's desk, and shows the positions of 
 the two projecting planes with the palm of his hand. Another 
 student designates with a finger the direction of the projecting 
 plane with the plane of the triangle. A third student criticises
 
 i8ij The Present Teaching of Mathematics in Germany 115 
 
 the result : " It is rather immaterial here whether the auxiliary 
 plane is chosen at right angles to the vertical plane or to the 
 horizontal plane.") 
 
 TEACHER: If it makes scarcely any difference, we will take 
 it otherwise than in the previous recitation. How did the assist- 
 ing plane lie there? 
 
 STUDENT : At right angles to the horizontal plane. 
 
 TEACHER: Then we will choose this time a vertical project- 
 ing plane. But first hold the paper triangle and the straight line 
 in the correct position. (The student does this.) Indicate now 
 the position of the auxiliary plane. (This is done.) Show the 
 intersection of the auxiliary plane and the triangular plane. (This 
 is also done.) We wish to determine the points which the cut- 
 ting-line has in common with the perimeter of the triangle. 
 Proceed. 
 
 STUDENT: Its vertical projection U" V" will fall on g". 
 
 TEACHER: Why? 
 
 STUDENT: Everything which lies in the auxiliary plane ap- 
 pears in the vertical projection as g". The intersecting line UV, 
 lying in this assisting plane, is also projected vertically in g". 
 
 TEACHER : Hence what points are now known to us ? 
 
 STUDENT: The points U" V" . 
 
 TEACHER: How are U" and V" found? 
 
 STUDENT : With the help of the proposition. The lower and 
 upper projections of a point lie on a common perpendicular to the 
 base-line. (/' lies on a | to base-line passing through U") 
 V is found in the same way as the intersections of B' C' with 
 the perpendicular to the base-line passing through V '. 
 
 TEACHER: Therefore what have we established? 
 
 STUDENT: We know the two projections of the cutting line 
 UV. 
 ' TEACHER: But what is our aim? 
 
 STUDENT: We wish to ascertain the intersection of g with 
 the plane of the triangle. 
 
 TEACHER : Then what remains to be done ? 
 
 STUDENT : We must make U V intersect with g. 
 
 TEACHER : How do we represent the intersection S of the two 
 straight lines?
 
 Ii6 Teachers College Record [182 
 
 STUDENT: We know, first of all, its horizontal projection S', 
 as the intersection of U' V and g. (Another student proceeds 
 at a sign from the teacher.) The upper elevation S" lies, in the 
 first place, on the perpendicular to the base-line through S', 
 and secondly on g". 
 
 TEACHER: How can we prove that we have constructed the 
 projection of 5 accurately? 
 
 STUDENT: We can also work out the construction with the 
 horizontal projecting-plane of g. 
 
 TEACHER: That would be too detailed for us. Who can sug- 
 gest something else? (Pause.) If S is really a point of the 
 triangular plane, so must, for example, B S lie wholly in the 
 plane of A B C. How can we prove that? 
 
 STUDENT: We must find out whether or not B S cuts the 
 side A C. 
 
 TEACHER: And how prove that? 
 
 STUDENT: B' S' determines on A' C' a point and B" S" on 
 A" C" another point. We must prove that these two points 
 lie on the same perpendicular to the base-line. 
 
 TEACHER: Prove it from our drawing. (A student does this 
 by making use of fine lines.) Reverse the blackboard. Now all 
 draw the construction just considered in your note-books. (While 
 this takes place, the teacher passes about the class-room from 
 one student to another. He finds a student who has made the 
 perpendicular to the base-line through U' cut A' B' instead of 
 A' C'. The teacher brings the mistake to the student's notice, 
 and in order to convince him, lets him turn back the blackboard 
 and run over the points in the construction, using again the 
 paper triangle as a model.) 
 
 TEACHER: Have you all completed the drawing? We must 
 also determine what parts of g appear as visible lines in the 
 vertical and horizontal projections and what parts do not so 
 appear. How can we determine this? 
 
 STUDENT : With the help of our paper triangle. 
 
 TEACHER: But we do not wish that. We will use the draw- 
 ing only. 
 
 STUDENT : Then we could perhaps proceed as we did recently 
 when we settled the proportion of visibility in the case of 
 warped lines.
 
 183] The Present Teaching of Mathematics in Germany 117 
 
 TEACHER: Who remembers that? Tell us how you would 
 here use the method mentioned. 
 
 STUDENT : I first seek the visibility in the vertical projection. 
 There is one and only one vertically projected straight line which 
 cuts g as well as A C in space. It is the straight line whose 
 vertical projection is the point U". I measure down this line, 
 and perpendicularly to the horizontal projection. The horizontal 
 projection shews me that I come first upon a point of g, then 
 upon a point of A C (namely C7). Therefore the vertically pro- 
 jected part of g that we are concerned with lies in front of the 
 triangular plane. I draw the projection of the part U" S" as a 
 visible line. 
 
 (Another student does the same thing with respect to the hori- 
 zontal projection.) 
 
 TEACHER: Draw that in your note-books. 
 
 (A signal bell announces the close of the hour.) You may 
 take that home with you. As your lesson for the next hour 
 solve the following exercise: A triangle is determined through 
 the co-ordinates of its vertices D (2, 3, 4), E (5, 5, 2), F (7, 
 2, 7). There is a straight line g determined by the two points 
 K (2, 6, 6) and L (7, i, i). Find the intersection of the straight 
 line g with the plane of the triangle D E F. 
 
 For the sake of clearness it is well that work with ruler and 
 compasses precede freehand drawing. The student should reach 
 the conclusion that the figures do not merely represent general 
 objects in space, as if they were drawn for the proof of geomet- 
 ric propositions, but that their results should conform to measure 
 as well as to shape. In the case of an especially gifted pupil 
 greater freedom may be allowed, even in the beginning. 
 
 As he advances, the student is encouraged to work out inde- 
 pendently more complex drawings. Many of these individual 
 productions show such ability to visualize objects in space that 
 one is quite justified in allowing the student to spend his time 
 on abstract, unpractical propositions. Such work often repays 
 the student a thousandfold. The author apparently enters a plea 
 for not attempting to eliminate all abstract knowledge in favor 
 of the concrete. 
 
 Opinions differ widely concerning the technical execution of
 
 Ii8 Teachers College Record [184 
 
 drawing. To quote from two writers : " The pupils make a 
 sketch in their copybooks, and as soon as they have completed 
 the construction it is redrawn on a clean sheet and filled in with 
 color. The pages must be handed in clean, and the drawings 
 must be exact, no bungling being allowed. If necessary, two or 
 even three copies are made." Again, " The drawing must be 
 correct. If this can be obtained with a lead pencil so much the 
 better. A laborious coloring of the work is a useless waste of 
 time, an amateurish nothingness." In many cases, indeed, the 
 students are allowed to produce technical, freehand drawings on 
 wrapping paper with lead and colored pencils. But, as Groth- 
 mann says, they do not train office draughtsmen. 
 
 Dr. Zuhlke considers as the best the happy medium. All 
 pupils should be allowed to work at the drawing-board, the 
 weakest and slowest only with pencil, the better and faster with 
 drawing pen, the most advanced being permitted the use of water 
 colors. 
 
 By some it has been suggested that colors be used to differen- 
 tiate that which is given in the problem from that which is 
 sought; also that the lines which are to assist in the proof 
 be made fainter than the head-lines. The author goes on to 
 state that the cherished plan of drawing the forward lines heavier 
 than the back, is, broadly speaking, a condemnation of the ex- 
 istence of parallel perspective. Moreover, it is not necessary 
 to make clearly-drawn figures indistinct by going over them 
 in colors. Ellipses do not need to resemble eggs. Neither does 
 the author advocate the use of the compasses in erecting a per- 
 pendicular to a line ; nor the laying of the ruler along the line 
 and the pushing of the triangle along it until the vertex falls on 
 the given point. The vertex of the triangle soon becomes worn 
 and the drawing consequently becomes inaccurate. In regard to 
 requiring the students to redraw twice or even three times, Dr. 
 Zuhlke advises the use of common sense and a little discretion. 
 
 THE USE OF MODELS 
 
 The use of models in descriptive geometry is discussed at some 
 length. There are many steps, it is said, between those who 
 expect the salvation of everything from the use of models, sep-
 
 185 J The Present Teaching of Mathematics in Germany 119 
 
 arating themselves from these aids at no step in the work, and 
 those who never use a model at all. By the use of such helps 
 the student is liable to lose his proper faculty of inner sight, the 
 building up of which is the aim of the course. Models are espe- 
 cially helpful in the beginning, while the student's powers of 
 space-conception are still weak, but it is recommended that they 
 be used with constant discretion, and that the student be not 
 permitted to have the model in hand for the entire work, but 
 only until the difficulties of presentation have been surmounted. 
 Miiller and Presler advocate a frugal use of models, saying that 
 the value of instruction in projection lies in the fact that one 
 learns to do without them. Professor Treutlein, too, is quoted 
 as saying : " The models are perhaps there to make themselves 
 superfluous." 
 
 In regard to the room assigned to drawing, conditions should 
 first conform to the requirements of school hygiene. For line- 
 drawing and descriptive geometry it is desirable to have a special 
 room. The room should be large enough to give to each student 
 four square meters of floor space. It should be placed in the 
 top story, and should have skylights facing the north as usual. 
 Since this kind of exact drawing strains the eyes more than 
 freehand drawing it should be given in the morning. This has 
 been accomplished in most schools. 
 
 THE TRAINING OF THE TEACHER 
 
 The method of procedure in the class room depends in an 
 important measure upon the more or less practical training of 
 the teacher. A drawing teacher not trained in mathematics will 
 give the exercise in line-drawing and descriptive geometry quite 
 differently from a teacher of mathematics unskilled in the rules 
 of drawing: whereas one will scarcely see a great difference 
 between a ready drawer with the capacity for mathematical 
 judgment, and a mathematician perfected in the rules of drawing. 
 In southern Germany descriptive geometry is universally 
 esteemed, and so it is natural that special work in the subject 
 is given in the training of the future teacher of mathematics. 
 
 In Bavaria, according to the regulations of May 26, 1873, for 
 examinations, every prospective teacher of mathematics was 
 examined in descriptive geometry, with the exception of its
 
 I2O Teachers College Record [186 
 
 application to perspective and shadow-construction. The present 
 order of examinations, regulated by the decree of January 21, 
 1895, requires a four-hour examination paper in descriptive 
 geometry. The student can acquire the requisite knowledge in 
 the technical colleges, which, since their foundation in the year 
 1868, are equally entitled with the university, to train the candi- 
 date for teaching mathematics and physics. In the university a 
 suitable course in descriptive geometry is amply provided for. 
 
 The situation in Wiirttemberg is similar, and in Baden it has 
 been looked upon for a long time as the duty of the university 
 to train the future teacher of mathematics in descriptive geom- 
 etry. In the University of Heidelberg lectures were given as 
 early as 1865 in descriptive geometry with respect to shadow- 
 construction and perspective. 
 
 Such also is the condition in central Germany. In Saxony 
 the subject was given in the Technical College of Dresden in 
 1870, and in the university in 1881, when Professor Klein was 
 called from the Technical College of Munich to Leipzig for the 
 newly created professorship in geometry. Since that time, Leip- 
 zig, as well as the southern universities, has given much atten- 
 tion to applied mathematics. 
 
 But in northern Germany the situation is quite different. In 
 the year 1880 W. Krumme impressively urged that careful in- 
 struction in descriptive geometry, as an integral part of the 
 mathematical training of the pupils of colleges, be considered, 
 and added the bitter complaint that the universities showed only 
 a slight comprehension of it. 
 
 There were in 1908, as assistants in descriptive geometry in 
 the Technical College at Berlin, six active teachers of the first 
 class of whom only one was an expert in applied mathematics 
 
 In the curriculum of almost every Gymnasium and Realschule 
 is to be found the subject of descriptive geometry, whereas in 
 the United States it is practically unheard of as a study in 
 the secondary schools. No doubt its prominence in the German 
 schools is partly due to the great manufacturing interests of 
 that country. Germany is far-sighted enough to recognize that 
 the development in her masses of such powers as that of being 
 able to visualize an object from its working drawing will have 
 great influence upon her industries.
 
 CHAPTER XVI 
 
 Eleanora T. Miller 
 
 It is well that Professor Smith at the close of his article has 
 added a word of encouragement to the American teachers, be- 
 cause we cannot read these reports without feeling a trifle de- 
 pressed by the apparent preponderance of evidence in favor of 
 German ideals and results as compared with the ideals which 
 prevail and the results which are achieved throughout the United 
 States as a whole. 
 
 American ideals of education are necessarily different from 
 those of Germany owing to the great difference in social condi- 
 tions, and it is our business to face our problems frankly and 
 courageously and to give all credit where it is due. There is 
 one matter in regard to the difference in the type of students 
 found in the secondary schools of the two countries which has 
 not been mentioned. In her secondary schools Germany educates 
 a more or less picked class of boys and girls while we are 
 attempting to educate the masses, to/ raise the great rank and 
 file of our young people to higher and higher levels of intelli- 
 gence. It is perfectly futile to try to transplant in their entirety 
 foreign educational ideas and methods to our own land. We 
 ma)' get suggestions from Germany but the applications must 
 be determined by our own national needs and our own national 
 character. One may learn much from studying the details of 
 the work of other countries and it is to be hoped that the reader 
 of these reports may find some helpful suggestions from a careful 
 study of the experiences of the German teachers of mathematics. 
 
 We are told that one of the advantages which the German 
 schools have over ours is in the length of the school day, and 
 we wonder whether this disadvantage might be overcome. Mod- 
 ern psychological experiments tend to prove that what we ordin- 
 
 187] 121
 
 122 Teachers College Record [188 
 
 arily think of as mental fatigue is little more than emotional 
 repugnance toward the work in hand, 1 and that if physical 
 conditions can be made ideal and the work can be made inter- 
 esting enough to appear to the student to be worth while, the 
 length of the school day can be increased without any danger 
 of impairing the health of the students. But, of course, physical 
 conditions are not ideal under our existing system, so there is 
 no use in our trying to do that which will continue to be im- 
 possible so long as we can not perfectly control our environment. 
 
 No one will deny that the training of the Gentian teachers is 
 vastly superior to that of our own; but we are gradually 
 raising the standards in this country. It is to be hoped that 
 the ideals of this generation will be realized by the next ; and 
 not the least important of these ideals is that we have a thor- 
 oughly trained and enthusiastic corps of teachers who are not 
 only willing but anxious to take advantage of every opportunity 
 that is offered to train to the highest point of efficiency the 
 scholar, the artisan, and, indeed, every individual that helps to 
 make up our complex social fabric. 
 
 It is not our purpose here to map out a definite course in 
 mathematics for secondary schools, but some suggestions ought 
 to be offered along the line of a course which will meet the 
 needs of the different classes of students, and Germany's experi- 
 ence can in some respects guide us. 
 
 To put all students through the same mill is neither practicable 
 nor possible, no matter how much we may wish to have every 
 student know just as much mathematics as we are able to teach 
 him during his high-school course. Our greatest difficulty is to 
 be met in the regular high school in the average city where we 
 must meet the needs of the boy or girl who is preparing for 
 college, the boy or girl who must drop out at the end of a year 
 or two, and the one who is undecided as to whether or not he 
 will continue work beyond that of the secondary school. 
 
 Supposing our present four-year course to continue, there 
 seems to be no reason why all students of the usual type of 
 high school should not be given the same kind of work for 
 two years, if the course is flexible enough to admit of a maximum 
 
 1 E. L. Thorndike, Mental Fatigue, Psychological Review, Nov. 1900.
 
 189] The Present Teaching of Mathematics in Germany 123 
 
 requirement for the capable student and a minimum requirement 
 for the slow one. The student can be made to feel that mathe- 
 matics is worth while if the subject is properly presented by a 
 thoroughly competent teacher, and this is all that is necessary to 
 hold his interest and call forth his best effort. This two years of 
 work should include algebra and plane geometry, and some trig- 
 onometry in connection with the subject of similarity of triangles. 
 Applications to business, home economics, mechanics, and the 
 various other sciences can be introduced in such a way as to 
 enlist the interest of both boys and girls of different bents of 
 mind. 
 
 The following two years' work might be required for those 
 who expect to continue the study of mathematics in some higher 
 institution, and elective for those who do not intend to go to 
 college but who have discovered their interest in the work pre- 
 viously done. As electives might be suggested a course in trig- 
 onometry ; a course in mechanical drawing combined with descrip- 
 tive geometry which should be a genuine mathematics course 
 and not a " snap " course in drawing, open to everybne ; a course 
 in solid geometry ; a course in advanced algebra ; and a course 
 in analytics and the rudiments of the calculus, provided there 
 were teachers trained to present these subjects properly. The 
 objection that the presentation of such subjects as analytics and 
 the calculus in the high school takes off the keen edge of the 
 student's interest in them when he goes to college is a reflection 
 upon the teacher's method, and is not an objection which need 
 be taken seriously. 
 
 Our vocational schools have problems which are more easily 
 solved because their needs are not so varied and their aims are 
 a little more definite. 
 
 The educational renaissance through which we are passing 
 to-day is being felt in the teaching of mathematics, and periods 
 of reform are always more or less disturbing. We are to-day 
 striving to make our concrete procedure come up to and fit 
 in with the needs which we have felt for some time, and to 
 adjust our mathematics to the new type of student that now 
 comes to the high school. When the readjustments are made, 
 however, and we settle down on the next rung of the ladder we
 
 124 Teachers College Record [190 
 
 shall probably find that the only very radical changes which have 
 been brought about are those affecting the teachers and their 
 methods, and that the subject matter will be much the same. 
 What we most desire is that our teachers of secondary mathe- 
 matics shall be thoroughly familiar with their field far beyond 
 the demands of the curriculum and that they be masters of 
 it on its historical, its practical, and its theoretical side.
 
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