di : 3A tpt bodpeadsy ty Ser aS Bea a ee ste haa () SB2sBes Paso IB PT O ) earning and Xabor. - & ! LIBRARY i University of Illinois. j & CLASS. BOOK. VOLUME. ; DUOMO 2. Sid eae f { Accession No. bop an a i Eee he A O _ Return this book on or before the Latest Date stamped below. A charge is made on all overdue books. U. of I. Library MAY pasa hd ig JAN 2 1M SEP 1 9 RECD 11148-S The Teaching of Mathematics tn the bigber Schools of Prussia The Ceaching of Mathematics in tbe higher Schools of Prussia BY J. W. A. YOUNG, PuD.. ASSISTANT PROFESSOR OF THE PEDAGOGY OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO LONGMANS, GREEN, AND CO. gt AND 93 FIFTH AVENUE, NEW YORK LONDON AND BOMBAY 1900 ATE NSE UB Bd Se a ee Vols Ty to Ve baieke bal ; 7 Ae ae et CopyRIGHT, goo, BY LONGMANS, GREEN, AND CO, — All rights reserved THE CAXTON PRESS oe NEW YORK. “ ; ‘ : 2 : wae x a 2H , ce ice. + ot Siders ee LIE 2300t-Y , Preface The well-known fact that the Prussians have long been studying the problems of education sys- tematically, thoroughly, and successfully, led me vecently to spend nearly an entire academic year in examining the outcome of their study as evinced in the present status of the work of education in Prussia. Some of the results of this examination °5 are presented herewith. It was my primary aim in examining the Prus- _ stan higher-school system to ascertain their meth- ods of teaching mathematics, but the work in mathematics cannot be understood without some acquaintance with the entire system of which the mathematical work ts a part and by whose spirit zt ts dominated. Consequently the following account combines a general sketch of the Prussian higher- school system with a more specific and detailed description of the work in mathematics. The reader who desires to learn more of the vi | Preface German higher schools in general than is contained in the condensed outline which suffices for the pur- poses of the present report, ts referred to the work of Russell} on the German higher schools. The material for the following account has been collected in part from the official and other publi- cations relative to these schools, and in part from many visits of observation to the institutions them- selves. The permit requisite for these visits was secured for me from the Minister of Education through the intermediation of His Excellency, the American Ambassador at Berlin, to whom I feel deeply grateful for this kindness. I met with the utmost courtesy in all my visits, the various Directors and instructors were most obliging and did all in their power to assist me in gaining the information of which I was in quest, and I wish to express my most hearty appreciation of their hospitable attentions. My thanks are also due to Professor H. Maschke, 1 Russell, German Higher Schools: The History, Organization and Methods of Secondary Education in Germany. Longmans, Green & Co., 1898, pp. 455. This work has come to my hands since the following pages were prepared, and I have read it with great interest. It is thorough and scholarly as well as interesting, and I know of no other work in English so complete, satisfac- tory, and reliable. Preface vii of the University of Chicago, formerly Professor in the Luisenstaddtische Gymnasium, Berlin, Mr. C. E. Linebarger, Instructor in the Lake View Fligh School, Chicago, and Mr. E. R. Breslich, a graduate of a Prussian gymnasium, for reading the following pages in manuscript, critically from their respective stand-points. F. W. A. Youne. Contents PAGE PREFACE, . . : : : : : 4 3 ny I. INTRODUCTION, : : ‘ ‘ : ae | Why study Prussian results, 1 A grave disparity, I Possibility of comparison, I The ratio of seven to four stated, 2 Questions raised, 3 What is to be learned, 3 II. THE GENERAL STATUS OF THE HIGHER SCHOOLS, 5 Common vs. higher schools, 5 Education a duty or a privilege, 6 The types of higher schools, 8 III, THE GOVERNMENT, The King, 9 The Minister of Education, 9 The provincial school-board, 9 Circular orders, 9 The Director, 11 The ordinary, 11 x Contents PAGE IV. THE TEACHERS, ; : e . ° . Ung Preparation, 13 Examination, 13 By whom conducted, 13 Requirements for admission, 14 The subjects, 14 The scope, 15 General, 15 Special attainments in mathematics, 15 For middle classes, 15 For upper classes, 16 The mode of the examination, 16 Written, 16 rai 3 Results of the examination, 17 The seminary year, 18 Purpose and nature, 18 The work, 18 Weekly conferences, 19 Observation of instruction, 20 Giving instruction, 20 The trial year, 21 Summary of preparation, 22 Appointment, 23 Ranks and honors, 23 Duties, 24 Salaries, 25 Pensions, 26 The ‘* jubilee,” 27 Total income, including estimated equivalent of pensions, 28 Purchasing power in America, 30 The effect of the pension system; tranquillity of life, 31 Contents xl PAGE eehete EOPILS,~ .«. : : : ° ° : SRK: Admission, 33 Classification and age, 33 VI. THE INSTITUTIONS, . a ee tear . ° sue a0 Classification, 36 Number, 37 Financial support, 38 Buildings, 38 Equipment, 39 Faculty-room, 39 Sessions, 40 The hourly pause, 41 VIL. THE CURRICULA, ; ‘ ° 4 : nA First curriculum, 43 Conference of 1890, 43 Detailed curriculum of gymnasium, 45 Synopsis of curricula of present century, 46 The curriculum in mathematics for the gymna- sium, 47 General aim of the instruction, 47 Scope of the instruction, 47 Methodic remarks, 50 The curriculum in mathematics for the Realgymna- sium and for the Oberrealschule, 51 German criticism of the methodic directions, 52 Distribution of the hours, 52 xil Contents PAGE VIII. THE INSTRUCTION IN MATHEMATICS, . ; «tae Classroom customs, 53 The size of the classes, 53 Teaching, not ‘‘ hearing recitations,” 54 Stress on the class-exercise, 54 ‘‘Socratic method,” 55 _ Oral work, 57 Diarium, 58 ‘« Chalk and talk,” 59 A lesson in algebra, 60 A lesson in geometry, 64 Objections considered, 67 The manner of the teachers, 68 Written exercises; neatness, 69 Number of teachers in mathematics during the course of a pupil, 70 Homogeneity of instruction, 70 The class-book, 71 Specimen entries, 72 Private study, 73 Specimen of allotment, 74 Text-books in mathematics, 75 Purpose of the text, 77 IX. THE EXAMINATIONS, ° ; ° ° ° e 79 Annual, 79 Final, 80 Written, 81 Setting the papers, 81 Independent work required, 82 Correcting the papers, $2 Determining the grade, 83 Oral, 83 Privileges of the graduate, 85 Contents Xill X. THE PROGRAMM, . ° . . ° A . 86 The scientific paper, 86 The school-report, 86 The curriculum, 86 The hours of instruction, 86 The class-work, 87 Examination papers in mathematics, 89 List of text-books, g2 Orders of the superior boards, 93 Specimen orders, 93 Chronicles of the institution, 95 Statistics, 96 Additions to libraries, laboratories, and muse- ums, 96 Stipends and funds, 96 Notices to parents and pupils, 97 XI. THE REFORMSCHULE, . i i : ; aoe Character and purpose, 98 Detailed curriculum of the Reformschule at Han- over, IOI Synopsis of various curricula, 102 XII. THE HIGHER EDUCATION OF WOMEN, . : ialOs No gymnasia for women, 103 What is done, 103 The quality of the work, 104 Progress being made, 105 XIII. CoMPARISON BETWEEN GERMAN AND AMERICAN WoRK, F ‘ 2 ‘ : : . 106 Basis of comparison, 106 Sketch of American course, 107 Work done compared, 108 Time ratios compared, 109 Comparative table of fractions of time given to math- ematics, 110 X1V Contents PAGE XIV. CONCLUSION, . : : : , . ‘ . 112 The disparity, how not caused, 113 Causes contributory to Prussian excellence, 114 Central legislation and supervision, 115 Expert government, I15 Other conditions, 116 A desideratum, 116 Uniformity of curricula, 117 Preparation of the curricula, 118 Supervision, 118 American conditions, 119 A serious loss, 119 Resolution IV. of the Committee on College Entrance Requirements, 120 What can be done now, 121 The preparation and status of the teachers, 122 Need of pedagogic training, 122 Raising of standards, 124 Enthusiasm and devotion needed, 126 The methods of instruction, 127 The stress laid on classroom work, 127 Work under instruction, 128 Resolution XIV. of the Committee on Col- lege Entrance Requirements, 129 More instruction needed, 130 The pause, 132 The distribution of the mathematical subjects in the course, 132 Three points of difference, 132 Scope of arithmetic, 133 Order of beginning, 134 Rate of continuing, 134 Gradual growth needed, 134 The method of instruction in mathematics, 136 The genetic method, 136 Class vs. individual, 137 The heuristic method, 138 Professor Miinsterberg’s views, 140 THE TEACHING OF MATHEMATICS IN PRUSSIA I TMntroduction The teaching of mathematics in the higher schools of Prussia deserves the serious atten- tion of those interested in the teach- Prieaad ing of mathematics in America, not oe only because it is the fruit of long labors by a nation that has stood and still stands in the forefront of educational progress, but also be- cause a comparison between the work accom. plished in mathematics in Prussia and in the United States reveals a disparity of a character so grave that American educators cannot afford to pass it by unheeded. There are very few subjects in which a com- parison of the quantity and quality of work ac- complished under different curricula pogsipitity of and methods can be instituted in any comparison. but the most general way. The result is af- fected by so many elements which cannot be I 2 Teaching of Mathematics in Prussia specified in black and white, or tested by exe aminations, that it is wellnigh impossible to find a satisfactory standard of comparison. In mathematics, however, this is the case to a much smaller extent than in other branches. The subject-matter of school mathematics has long since been so systematized, and its nature permits so little variation in the topics taken up and the order of their consideration, that the quantity of work done may be quite clearly de- scribed by a list of topics and its quality suf- ficiently well tested by examinations. The present study of the Prussian secondary school system was begun under the impression Ratio of (quite current in America) that, while seven to four. the work of the Germans is perhaps more thorough, it is accomplished with a great- er outlay of time than is devoted to the same subject-matter in this country. But compari- son of curricula and time-schedules reveals the startling fact (which will be substantiated in de- tail in the sequel) that zz the work in mathematics done in the nine years from the age of nine on, we Americans accomplish no more than the Prussians, while we give to this work about seven-fourth (1.72) tomes as large a fraction of the total time of in- struction as do the Prussians. In other words, the Prussians give about 1.2 years of the nine years in question to mathematics, accomplishing fully as much as the Americans, who Tntroduction 3 give about 2.1 years to the same work—a differ- ence of nine-tenths of a year, or one-tenth of the total time of instruction in these nine years. This state of affairs is certainly one which demands most careful consideration at our hands. Is there really so great adis- questions for parity as appears on the face of the ‘consideration, time-schedules? Is there in reality any disparity at all? Ifso, to whatisit due? Is it possible for Americans so to modify their system and methods as to diminish the disparity? What lessons can Americans learn from the Prussian system? These and other questions suggest themselves at once, and it is with these ques- tions in the foreground that the following ac- count of the Prussian system should be read. The writer wishes to say at the outset that, while he believes that there is a real disparity and that it ought to be lessened, he whatisto by no means advocates that the Prus- __ be learned. sian system as such be adopted or imitated here. It does seem, however, that the indisputable su- periority which the facts mentioned above show the Prussian system to have in its own environ- ment, over the American system in its environ- ment must oblige American educators to study the Prussian system most carefully, especially along its lines of divergence from their own. While the outcome of such study may be that little or nothing is found which we can directly 4 Teaching of Mathematics in Prussia adopt, hints may perhaps be gleaned which we may adap? to our own circumstances with signal profit. Education is more a problem of human- ity than of nationality, and while distinctively German methods might not prove strong else- where, those results which the Germans have attained as men and not as Germans must be of great significance the world over. I] The General Status of the higber Schools The distinction between the common and the higher schools (Volks- und hihere Schulen) must be noticed at the very outset. Thisis (| not a distinction in any way analo- vs. nigher gous to our grades and high schools, *"°'* but constitutes a complete differentiation of the boys from almost or quite the beginning of their school career into two distinct classes. In the Volksschulen the aim is to train good and faithful citizens ; the process is called Erziehung (“bringing up”). In the higher schools, on the other hand, the aim is to impart learning and to turn out men who are educated or cult- ured (gedz/det); the process is called Unterricht (instruction) and leads to privileges and respon- sibilities before the civil and the military law, and the unwritten social law as well. The higher schools proper take the boys at the age of nine and have a curriculum covering nine years ; in many cases a preparatory school with a course of three years is connected with the institution, so that a boy of six years may step 5 6 Teaching of Mathematics in Prussia into the work of a higher school. As the com- mon schools do very little in mathematics, only the work of the higher schools will be con- sidered in what follows. Several American writers have already given general descrip- tions of the German common school system, to which the reader who may be interested in the work of these schools is referred.’ A thorough German compend covering the entire educational system is that of Peter- silie,? in which all grades of institutions from the Universities down are described and their regulations collated. This is done in consider- able detail for Germany, and in a summary manner for the other principal countries of Europe. In Prussia the state regards attendance upon a higher school as a previlege ; for the common Education 00d the state may restrict the num- adutyora ber of persons admitted to such at- privilege. tendance. But attendance upon the common schools is regarded as the duty of those not having better opportunities, and is enforced by the state. The attendance upon 1 For example, Klemm, Zuvopean Schools, D. Appleton & Co., 1889; Seeley, Zhe Common Schools of Germany, Kellogg & Co., 1896. ® Petersilie, Das offentliche Unterrichtswesen im deutschen Reiche und in den tibrigen europdischen Kulturlindern, Leipzig, 1897, 2 Bde., pp. 448, 608. The General Status of the thigher Schools 7 higher schools is now being restricted by the state by the simple expedient of founding fewer new institutions than would be adequate to meet the present demand for admittance, which is far in excess of the number that can be re- ceived by the institutions now in existence. This is done to abate the crying evil which Bismarck called the Adzturzentenproletariat (“beggar-graduates”). The graduate of a higher school is admitted to occupations closed to all others, and the social usages are such as to prevent him, on pain of losing caste, from entering any of another large group of occupa- tions. Consequently the occupations which are considered suitable for graduates are terribly overcrowded, and since the pressure cannot be relieved by overflow into other occupations, it must be relieved at the source of supply. The condition of affairs is graphically illustrated by an experience of an American resident of Ber- lin. An educated German called in silk hat and gloves to deg. “Why don’t you work?” asked the common-sense American, and was met with the indignant reply, “ That would not be in keeping with my social station” (“ Das ware nicht standesgemiss’’). The higher schools are divided into three types: the Gymnasium, with both Latin and Greek; the Realgymnasium, with Latin but no Greek; and the Oderrealschule, with neither 8 Teaching of Mathematics in Prussia Latin nor Greek. The characteristics of these institutions will be described later on, this Thetypes Prief mention sufficing for the present. ofhigher The corresponding work in Amer- re aes ica is divided between institutions of differing character, and there are here no single institutions analogous to the German higher schools; we shall therefore be obliged to retain the German names in speaking of these schools. III The Government The primary source of educational authority in Prussia is the king. The present king (Em- peror William II.) takes an active in- he govern- terest in the work and was himself _ ing bodies. educated in a gymnasium. The head of the actual educational work is the Minister of Spir- itual, Educational, and Medicinal Affairs (Gezst- licher, Unterrichts- und Medtzinal-Angelegenhet- ten), who is appointed by the king and is a member of the royal cabinet. He is assisted by over twenty active councillors for educa- tional matters (vortragende Rathe). The Minis- ter, in turn, appoints a school-board (Provin- gtal-Schulkollegium) for each of the thirteen provinces of Prussia, to whom the detailed su- pervision of the schools is intrusted. Besides particular communications to the separate institutions, both the Ministry and the provincial boards issue frequent Circular circular orders or general bulletins oreerss (Circularver figungen) to the institutions respec- tively under their charge. These take up on 9 10 Teaching of Matbematics in Prussia occasion matters of pedagogic method, and of administration even in detail, and thus tend to produce great uniformity in the work of all the institutions of the kingdom. The orders of in- terest to the general public are usually pub- lished in the annual announcement (Programm) of the school, and in treating of the latter, cita- tions will be made. All the orders since the beginning of the century which are still in force have been col- lected in the work of Wiese-Kiibler,! which is officially recognized, and upon which some of the statements of this paper are based. The orders are well grouped and indexed, and the work as a whole constitutes a complete and authoritative exposition of the organization and regulations of the Prussian higher school system. The provincial school-board is composed of picked men of experience in school-work, who The provine Gevote their entire time to the duties cialschool- of this position and who receive the see) highest salary paid. Each member has a number of institutions assigned to him for special personal supervision; he keeps in close touch with each, and informed as to its work by personal communication with the Di- 1 Wiese-Kiibler, Verordnungen und Gesetze fiir die hohere Schulen in Preussen, 3te Aufl., Bd. I., 1886, pp. 488; Bd. IL., 1888, pp. 521. The Government if rector, by visitations (for inspection of instruc. tion and conferences with the teachers), and by conducting examinations. He is the connect- ing link between the school-board as a whole and the institutions under his especial charge. The Schulkollegium of the province Branden- burg, in which Berlin lies, has commodious and well-fitted quarters (three floors of a large building) and a considerable clerical force. The head of the institution is the Director, appointed by the provincial school-board with royal approval. He administers the ne schools affairs of the institution, assigns to eee each teacher his work, and appoints an “ ordi- nary” (Ordinarius) for each class. These are, as it were, subdirectors supervising the work of single classes as the director does that of the entire institution. The ordinary is usually the teacher who gives the largest number of hours of instruction to the class, and consequently the teachers of Latin are sure to be among those who are called on to perform this duty. The ordinary has charge of all the routine super- vision of the class as a whole, and comes into closer personal contact with the pupils than do the other teachers. Everything not strictly in the usual course of the work of instruction of a class must be referred to the ordinary. Every question not 12 Teaching of Mathematics in Prussia strictly in the usual course of administration of a single class must be referred to the director, and so on, each officer referring to a higher au- thority all questions not falling within the scope of his own well-defined powers. IV The Teachers The first step toward becoming a teacher in the higher schools of Prussia is the acquisition of a liberal education and of sufficient scientific attainments in the subject which the candidate wishes to teach. It is requisite that he have completed the course of a gymnasium (for certain subjects that of a realgymnasium will suffice), and that he have studied three years in German Universities. The adequacy of his preparation is tested by an examination. This is conducted by a board (Kénigliche wts- senschaftliche Priifungscommission), appointed for that purpose by the Minister of Edu- the exam. cation. There are ten such boards in pb the kingdom, one board serving for two prov- inces in a few cases. Their seat is always in a University town, and their membership is made up almost entirely of University professors. This examination is known as the examina- tion pro facultate docendi, or the “ Staatsexa- men.” To be admitted to it the candidate must submit Preparation. ee 14 Teaching of Mathematics in Prussia a. A certificate of maturity (equivalent to our diploma of graduation) from a German gymnasium; if the principal subjects (see be- low) are taken from the following: Mathemat- ics, natural sciences, foreign modern languages, the candidate may offer a certificate of ma- turity from a realgymnasium. 6. Documents to show that he has studied three years in a German University (of these at least one and one-half years in a Prussian University) ; if one of the subjects is English or French, the candidate may by special permis- sion replace one year’s University study by study of the language in question in an institu- tion or ina country in which the language is spoken. The candidate specifies the particular sub- jects in which he seeks authorization to teach and the grade in which he wishes to obtain the teacher’s certificate. There are three grades—lower, middle, and upper— each constituting three years of the nine years’ course. He must offer at least two principal and two subordinate subjects. These subjects are selected from a list which is practically that of the subjects taught in the gymnasia (see cur- ricula below), the combinations being subject to a few restrictions which are of little conse- quence in the present paper. The scope of the examination is twofold, The subjects. The Teachers 15 testing the candidate’s fitness for the post of teacher in general and in particular. The spe- cific requirements are: 1. General.—All candidates are examined in philosophy and pedagogy, the German lan- guage and literature, and, if Chris- tibet: tians, in the contents of Holy Script- _ theexami- ure, Church history, and the dogmas aa of that church (State Church or Roman Catho- lic) to which they belong. The object of this part of the examination is to determine whether the candidate possesses that general culture which is to be demanded of all instructors in higher schools. The examination is rigorous, and as it may take up topics from a wide field, and as the ex- aminer in each subject is usually a University professor and always a specialist in the subject, the candidates often anticipate this examina- tion with more apprehension than that in the special subjects which they wish to teach. 2. Special Attainments.—To obtain the teach- er’s certificate (Oberlehrerzeugniss), the candi- date must obtain the authorization (Befahigung) to teach two (principal) subjects in all classes and two (subordinate) subjects in the middle classes. For the subject of mathematics the scope of the examination is as follows: a. For Middle Classes.—Plane and solid geom- etry, algebra through quadratics, logarithms, 16 Teaching of Matbematics in Prussia properties of the decimal system of numeration, equations of the third and the fourth degree, spherical trigonometry with applications to mathematical geography, plane analytic geom- etry, and the elements of the differential and integral calculus. b. For Upper Classes.—In addition to the fore- going the candidate must show that he possesses such acquaintance with the most important branches of higher geometry, analysis, and an- alytic mechanics as will enable him to treat in- dependently a not too difficult problem from one of these fields, and he must be acquainted with the more important literature of these subjects. The examination consists of two parts: the written examination, in which the candidate The mode of Prepares papers privately on assigned examination. tonics, and the oral examination, in which he appears before the Commission in person and may be examined at will by each member. » I. The Written Examination.—Subjects are assigned to the candidate, one from each of his principal subjects, also one from any subordi- nate subject in which he may wish to obtain the authorization to teach in all classes, and one from philosophy or pedagogy. Not more than three subjects may be assigned altogether. The candidate prepares at home a paper on each The Teachers . 17 subject assigned. Those from classical phi- lology are treated in Latin, those from foreign modern languages in the language concerned; all others in German, except by special permis- sion. Eight weeks’ time is allowed for the prep- aration of each paper (an additional eight weeks may be granted upon due application, and still more if necessary), and at the close of the total time accruing for all subjects all the papers are handed in, with the candidate’s as- surance that they were prepared by himself with no other assistance from persons or books than that specified in detail by him. If the candidate submits a printed paper written by himself, it may be accepted provided the subject and contents are satisfactory to the board. If the paper have been approved by the faculty of a Prussian University as a dis- sertation for the Doctor’s degree, only the sub- ject of the paper is scrutinized by the examin- ing board. 2. The Oral Examination.—If the results of the written examination have been sufficiently good, the candidate is notified to appear for the oral examination which extends over all the subjects offered by him, and includes the gen- eral culture topics as well. Results of the Examination.—The candidate may be passed unconditionally, conditionally, 18 Teaching of Mathematics in Prussia © or rejected. In appropriate cases there are open to him repetition of the examination, supplementary examinations (to work off con- ditions), and additional examinations (in new subjects for extension of teaching privileges). After it has been ascertained by the examina- tion that the candidate is possessed of liberal Thesemi. Culture and of sufficient specific sci- nary year. entific attainments in the subjects he wishes to teach, he must next devote a year (Seminarjahr) to the study of the art of teach- ing with a view to the practical exercise of his profession. To give opportunity for this study, pedagogic seminaries have recently been organized in connection with various ones of the schools, and to these the candidates are assigned in numbers not to exceed six for each semi- nary. The director of the institution conducts the work of the seminary, and the seminaries have been located in institutions whose direc- tors are men eminent in the pedagogic world. The work of the seminarist is of three sorts: 1. Weekly conferences of the seminary. 2. Observation of teaching. 3. Teaching under supervision and guid- ance. The weekly conference is held under the presidency of the director, and an experienced The Teachers 19 professor of the subject which is at the time under special consideration also attends the ses- sion and participatesinits work. The the weekty Gueteines) are varied at the discre- conference. tion of the director and include informal talks and lectures by him, papers by the members on assigned topics, reports by the members on the instruction they have witnessed ; in the case of members of the faculty, this report is confined to a statement of the facts observed, while in the case of colleagues in the seminary, the method of teaching may also be discussed and suggestions for improvement made. The exer- cises are not very formal, and questions and discussions may constitute an important feat- ure of the proceedings. The seminarists are also guided in the reading of pedagogic litera- ture. The aim is to shape the work of the seminary so as to cover the most important topics dur- ing the course of the year; such as the consti- tution of the higher school system, the curric- ula of the schools, the aim of the work as a whole and of its various parts, the methods of teaching in general, more detailed considera- tion of the teaching of those subjects which the members of the seminary expect to teach, the subject-matter of these latter subjects from the teacher’s stand-point, text-books and other aids and appliances, etc. 20 Teaching of Mathematics in Prussia Immediately on entering upon his work, the seminarist is set at observing the instruction Observation Which is being imparted throughout ofinstruction. the institution. About twelve hours per week are given to these visits of observa- tion which at first range over all subjects and from the lowest to the highest class, in order that the seminarist may understand the scope of the work as a whole and the interrelation of its parts. Later, his visits are concentrated more upon classes in the subjects which he is preparing to teach, and finally upon a class which is soon to be put under his own instruc- tion fora time. As already mentioned, he pre- sents a report of his observations to the weekly conference, and his report receives the criti- cism of his colleagues and of the experienced teachers who may be present. The seminarist also attends the regular fac- ulty meetings, but he has no voice in the dis- cussions except when called upon to report concerning pupils under his charge. When the seminarist has thus visited a desig- nated class for a sufficient length of time to be- actual come familiar with the character and instruction. methods of the work being done, he is permitted to give the instruction himself under the direction and supervision of the per- manent teacher of the class. The entire re- sponsibility for the work done still rests upon The Teachers 21 the latter, who is usually present during the hour, though less frequently as the candidate progresses satisfactorily. The director and other candidates also often look on. On occa- sion, the teachers present correct the candidate in the class-room, point out defects or suggest better methods, and he is supposed to take coun- sel with them privately as to his work. The other seminarists report on his teaching and criticise it freely at the weekly conference, and the teachers who saw him at work add such remarks as they may deem wise. The candi- date teaches only a few weeks in any one class, when he is assigned to similar work in another class, his assignments being at first to lower classes, and later to the higher classes. In the seminary year the candidate is entitled to no remuneration, though there are some small stipends. At the close of the year the director submits to the provincial school-board a full report on the work of the seminaries; if the work of any has been unsatisfactory, he is debarred from the opportunity of proceeding farther in the course of preparation for the profession of teaching, while the others are advanced by the board to the ¢rial year. This year (Probejahr) is usually passed in a different institution from that in which the sem- inary year was spent, and in it the candidatus 22 Teaching of Matbematics tn Prussia probandus is given six to eight hours per week of instruction to do, and the classes are placed under his charge as their regular in- structor, though he still works under the supervision and guidance of his superiors. In some cases the exigencies of instruction make it necessary to assign him more instruction to give than that mentioned above. He is paid by the hour for what teaching he does. Prior to the institution of the seminary year (1890), the trial year included work of the character now done in the seminary year. We have thus seen that, apart from the time requisite to pass the extended and searching Summary of €Xamination pro facultate docendt, a preparation. minimum of five years of special prep- aration is required of everyone who would become eligible to appointment as teacher ina Prussian higher school. Three of these years — are devoted to preparation in subject-matter and two to learning the art of teaching; in the first of the latter, theoretic study of pedagogic problems and methods preponderates, while in the second the candidate tentatively begins the practice of independent teaching. There is small wonder that few of those who survive all of these tests prove later to be poor teachers, and that, on the whole, German teaching leads the world. Having passed through the seminary year The trial year. The Teachers 23 and the trial year satisfactorily, the candidate is eligible to appointment as teacher. This may be either provisionally as assistant The ati teacher (wzssenschaftlicher Hiilfsleh- ointment. rer), Subject to dismissal upon three months’ notice, or definitely as instructor (Oderlehrer), removable only for serious offences and after formal trial.! In the interests of the work instructors may be a. Transferred to other positions of not lower rank and not less pay, with payment of moving expenses ; 6. Placed temporarily upon the inactive list with prescribed pay; or c. Placed permanently upon the retired list (Ruhestand), with corresponding pension (see below). The Minister grants the title Professor to in- structors who have evidenced scientific or ped- agogic excellence,and upon the same Ranks and grounds professors may receive the ponares rank of Councillors of the Fourth Class (Kathe vierter Klasse). This gives them social equality with the Councillors of the Fourth Class in the other branches of the government, with men of 1 In certain urgent cases instructors are suspended or forbidden to exercise their functions. They receive half pay while suspended, and the other half is also paid to them if upon trial they are found innocent. 24 Teaching of Ahatbematics in Prussia high rank in’ other professions, who are Coun- cillors of the Fourth Class, such as University professors, and the like, and gives them social precedence of such as have not this distinction. The conferring of these titles has no effect upon the salary of the recipient. The appointed teacher takes a prescribed oath of office by which he becomes an official of the state. As such he must culti- . vate loyalty to the king and the realm, advance the interests of education and of his institution as far as he can, and in particular give instruction not to exceed the following number of hours per week: Director, 14-16. Instructor, 20-22. The “hours” are at most fifty minutes, and at the close of each hour there is a pause of at least ten minutes. The maximum number of hours may be required only if the classes are small and if the subject of instruction does not demand from the teacher a time-consuming correction of papers. The teacher also is to do without extra pay such emergency teaching as may be rendered necessary by the death of a teacher of the insti- tution and also to replace free of charge such of his colleagues as may be absent through ill- ness, through leave of absence on the ground of ill-health, through being called into military Duties. The Teachers 25 service, or through jury duty. If ateacher ob- tains leave of absence for any cause or other than one of these mentioned, he must himself defray the cost of filling his place during his absence. If a teacher is obliged to travel for his health, to visit springs or baths, an appro- priation in addition to his regular salary may be granted him in view of the extra expense to which he is subject. The salary of an instructor is $648 per an- num (1 Mark = 24 cents) during the first three years, at the close of which time he re- ceives an increase of $72 per annum, and each three years thereafter a like increase is made until in twenty-four years the maximum salary of $1,224 is reached. One-half of the in- structors in the complete institutions (see be- low) and one-fourth of those in the incomplete institutions receive annually a fixed addition to their salary (feste Zulage) of $216. This addi- tion when once attained is permanent and is usu- ally granted new recipients in order of senior- ity as vacancies occur (by death or transfer to the inactive or retired list). Instructors also receive an appropriation for rent varying with the population of the town in which the insti- tution is located. This appropriation ranges from $216 for Berlin to $86 in the smallest vil- lages. This is the only item in which the in- comes of instructors vary according to the The salaries. 26 Teaching of Mathematics in Prussia location of their institutions. Consequently the incomes of instructors will gradually in- crease in Berlin from $864 to $1,656, and in the smallest villages from $734 to $1,526, while other cities and towns have ranges intermedi- ate between these. After ten years of service instructors are en- titled to pension in case they are permanently Thepen- Gisqualified for teaching by physical sions. _or mental weakness or by incapacity for the work. The amount of the pension de- pends upon the number of years which the teacher has served, and also upon the income which he is receiving at the time of retirement. The entire income of the instructor at the time (salary, rent allowance, and fixed addition, if he is the recipient thereof) is the base upon which the pension is computed. After ten years of service the pension is one-fourth of the base, and for each additional year of service one-six- tieth is added to this fraction, until after forty years of service the maximum pension of three- fourths of the base may be received. In count- ing years of service the seminary year, the trial year, and military service performed after the age of twenty are included. Instructors are not required to contribute to a pension fund. In Berlin the amount of pension would ac- cordingly range from about $234 at the end of The Teachers 29 ten years’ service to $1,227 at the end of forty years of service. After forty years’ service instructors are privileged to retire upon pension even though not incapacitated, but this is rarely The«jupie done. Inthe programme of one in- lee.’’ stitution a long list of instructors was pub- lished who have taught in that institution up to their “Jubilee” (fifty years of service). Each of the older institutions could doubtless furnish a more or less extended list of this kind. "The completion of fifty years of service is usually celebrated appropriately. The veteran “ Fubz- lar’ often receives letters and tokens of dis- tinction from the government, addresses in rec- ognition of his long and valued services from his colleagues, and tributes of respect from his pupils, present and past. The deference paid to age and experience in the German school-world is very marked. The responsible posts, the directorships, the portions of the work of instruction considered the more important or desirable, are usually allotted to elderly or old men. The younger men work with them in cheerful subordination and with genuine respect for the greater wisdom of greater experience. If we wish to consider the total professional income of the Prussian teacher, we must take the assurance of pension into account. Pensions 28 Teaching of Mathematics tn Prussia can be purchased in life insurance companies in Berlin, by the payment of a fixed sum an- Total profess NUAally for a term of years, and the sionaline study of their rates may enable us to aeok form an estimate of the market value in cash of the assurance of pension which the Prussian teacher enjoys. The conditions on which the pensions are paid by the companies differ somewhat from those of the schools as outlined above, otherwise the rates of the com- panies might be taken as being the cash value of the assurance of pension. The following table is constructed from the prospectus of a strong company doing business of this sort. If payments are begun at the age of twenty- five, the annual payment of P dollars until the age A is attained would secure at the age A a lifelong pension equal to that which the Prus- sian teacher would receive if retired upon pen- sion at the age A. Awssagess 35°./40.5 45 9.50 ° 55 s0omuaoe Beat alelan S $366 $300 $243 $242 $197 $142 $94 NOTES 1. Different teachers retiring at the same age A might not re- ceive precisely the same amount of pension owing to difference in the ages at which they began service as teachers and the age at which they attained the fixed addition, The range of variation would not be very considerable ; in the above table an estimate of the average has been used. 2. In making the table it was assumed that the fixed addition is The Teachers 29 attained between the ages of forty-five and fifty years. The teach- er’s pension at the latter age would thus be quite considerably larger than at the former, which accounts for the very slight de- crease in the amount of the annual payment P, requisite for the purchase of the same pensions to take effect at these two ages. The conditions of assurance correspond quite closely with those to which the teachers’ pen- sions are subject, with one important exception —viz., the payment of pension by the company begins at a stipulated date, neither sooner nor later, and the state of health of the assured or his need of the pension has no influence what- ever on his receipt of it. The annual payment of $351 for a period of ten years assures defi- nitely that payment of pensions will be made from the age of thirty-five on, while the teacher receives pension at thirty-five only if perma- nently incapacitated for work. Likewise the annual payment to the company of $197, for instance, assures the payment of pension at the age of fifty-five, dut not before, no matter how urgently it may be needed. The teacher, on the other hand, has the continual guarantee that he will receive pension whenever he may need it. It is not easy to deduce from the table above the annual payment which should be made to secure precisely the same guarantees which the school pension system gives. A very rough approximation may perhaps be obtained by re- garding the average of all the rates—v7zz., $224— 30 Teaching of Matbematics tn Prussia as expressing the amount of the mean annual payment sought. Assuming this as the average annual cash value of the pension guarantees, the incomes of the Prussian teachers when compared with those of non-pensioned teachers should be considered as ranging in gradual ascent, sure to each teacher, from $1,088 to $1,880. To compare these salaries with American salaries, the difference in the purchasing power Purchasing ©! Money in the two countries must bower in be taken into account. Conservative America. : estimates made by those who have lived for extended periods of time as settled residents in each country regard the purchas- ing power of money in Germany as about four. third times that in America. If this be correct, the German incomes would be equivalent in value under American circumstances to a range from $1,451 to $2,507 per annum. The American work in mathematics corre- sponding to that for which these salaries are paid in Germany is done in the grades below the high school (five years), in the high school (three years), and in the freshman year in col- lege, the younger teachers receiving the small- er salaries and doing the work of the earlier years. The above is a crude attempt to estimate the value of the pension guarantees, but by far the The Teachers 31 most beneficial effects of the Prussian pension and salary system are not capable of inclusion inacash estimate. The Germans ap- Tranquitity preciate that the teacher can do his oe. best only in an atmosphere of financial and men- tal tranquillity. He must himself be continually growing, and if he is embarrassed by financial cares and harassed by struggles to improve his material position, his growth is retarded and the quality of his work inevitably deterio- rates. The teacher is spurred on to his highest achievements by devotion to his calling and by the inspiration of seeking, finding, and impart- ing truth, but not by the competition of the mart or by the goad of necessity. The educational system of Prussia recognizes this truth, and while insisting on high standards and severe tests at the outset, assures a tranquil career to those who have given evidence of their fitness. The German teacher works with a sense of se- curity ; security in his position without regard to the occurrences of politics or the whims of the powerful and the influential; security ina modest competency while at work; security in case of a “rainy day;” security in his profes- sion as a lifework, in the assurance that if, as a young man, he maps out for himself a pro- gramme of work, study, and research requiring decades for its completion, nothing but the flag. 32 Teaching of Mathematics in Prussia ging of his own assiduity or the collapse of his physical or mental powers will prevent its suc- cessful execution; security, finally, that after a quiet life of patient, undistracted, fruitful toil in his noble vocation he will reach a well-cared for and honored old age, with all needed repose from his well-finished labors. V The Pupils Boys who apply for admission must have completed their ninth year, must be able to read German in both German and Latin characters, have a clear and legible handwriting both in German and Latin script, be able to write from dictation without bad orthographic errors, know a little biblical history, ard be familiar with the four fundamen- tal overations on whole numbers. The pupils are classified into nine classes, each with a course of one year. The following table gives the names of these classes, ctassification the abbreviations which we shall use 9"448¢ for them, and the average age of the boys in each near the close of the year. In forming this table, the writer examined statistics as to the average age of pupils in twenty-three insti- tutions. These institutions were selected so that all kinds were included and different loca- tions represented. The variations in the aver- age ages as between institutions were found to 33 Admission. 34 Teaching of Mathematics in Prussia be so slight that these twenty-three institutions were regarded as giving a correct idea of the range of ages. The average age for each class is published by every institution. The average of these averages constitutes the average of the table. The highest and the lowest average found for each class is also given: Class. Oberprima. Unterprima Obersecunda. Untersecunda. Obertertia. Untertertia. Abbreviation...| IA. || IB. | ITA. | IIB. | LITA.) TITB.) TV. even ae Average age. ..| 19.4 | 18.4 | 17.4 | 16.5 | 15.4 | 14.3 | 13.2 | 1r.8 | 10.7 Highest aver- age age...... 20:9 | 19,4 | 18.6 | 17.2 | 25.97| 15.5 [23.0 Mere gare Lowest aver- ape age... ss. 18.2) }.27.5 ] 16.3 | 25.8 24.5 1523.5) (era, 0 ek Sars Lowest age possible for any single pupil, about..] 17.9 | 16.9 | 15.9 | 14.9 | 13.9 | 12.9 | 11.9 | 10.9] 9.9 The table shows that the variation in age among the pupils of any one class cannot be very great, and the appearance of the classes themselves confirms this. The class-names given to the pupils are formed by adding zer to the names of the classes: as Primaner, Sextaner. The teachers of each class, in conference, name the first doy or Primus of the class. The appointment is made on the basis of scholarship The Pupils 35 and is considered a high honor. The Przmus acts as monitor, supervises the room in the ab- sence of the teacher, keeps the class record, and says prayer at the beginning and the end of the day’s session. VI The WMnstitutions The institutions are divided, according to the character of the work and the length of the course, into the following six classes ‘(the customary abbreviations being given in parentheses): Classification A. Nine years’ course. Gymnasium (G), has both Latin and Greek. Realgymnasium (RG), has Latin but no Greek. . Oberrealschule (ORS), has neither Latin nor Greek. B. Six years’ course. Progymnasium (PG). Realprogymnasium (RPG). Realschule (RSch). The institutions under B do precisely the first six years’ work of the corresponding institu- tions with nine years’ course. The institutions of Class A are called “ Com- plete Institutions” (Vol/anstalten), and those of 36 The Unstitutions a7 Class B “Incomplete Institutions” (Vichtvollan- stalten). The number of institutions of each kind in Prussia was in 1896 as follows: Pe IMIASIETE Pulte Ses '4)s/d) ee os e\ e's! 273 PGA TEINASIOI 0.1. )r e's! o'e ol ob! e al 86 Mierreaschulene yc e tea 24 POMS GL senior aul) ig 45 Pea LOD WV INIMASIONy 1121) cele 9 s'¢e 71 | Cie EST Con bck Sk an Ue nS 73 Ue hte aa weet te rcs ce a 572 Another classification: Witiewatinand Crreeki.J..i.1. 5 5318 With Latin but no Greek. ....157 With neither Latin nor Greek. 97 We have already stated that these institu- tions do not correspond closely to any kind of American school, and that hence the German names will be retained in speaking of them. The ideals, administration, and discipline are throughout strictly those of the school, but the curriculum extends somewhat beyond that of our best high-schools. In mathematics, the work done covers approximately the same 38 Teaching of Mathematics in Prussia ground as our course to the close of the fresh- man year in college. In addition to their income from tuition fees (about $33 per annum per pupil),and from en- Financial dowments (usually not large), insti- sgt ie tutions may be supported by the state, by the city in which they are located, by private organizations, or by private individuals. The state makes good deficiencies in the budget of institutions not primarily supported by it, and all institutions, even those receiving no support from the state, are obliged to conform to the same curricula and are subject to the same in- spection and control by the educational authori- ties. The school-buildings are usually located in the interior of blocks, away from the noise and The build: bustle of the street. In Berlin they weer are as arule modern and well adapted to their purpose, those of the last few decades especially being models of school architecture. There is a strange deficiency in all the build- ings, even the newest, in two respects: only one blackboard, from five to eight feet in length, is to be found in each room, and the seating arrangements are not good, usually consist- ing of desk-benches holding from four to six pupils. When called to the board the older pupils pass before the others of the same row, while in the lower classes the boys often run Tbe Tnstitutions 39 along back of the others on the benches on which they are seated. There isalwaysa large hall for physical culture (Zurnhalle, gymnasium in our sense of the word), fitted out with appa- ratus, and a yard for open-air exercise whose area (roughly estimated) is at the minimum about equal to that covered by the buildings and at the maximum is several times as great. Each institution has two good libraries, one for the pupils and the other for the teachers. Catalogues of the first are sometimes The equip- published indicating the classes to ment. which each book is suited. The second con- tains important scientific and pedagogic works, and both libraries are increased each year by purchase and gift, the titles of the new acquisi- tions being mentioned each year in the Pro- gramm. The physical and the chemical labora- tories and the museums of natural history seemed to be good, but the purpose of the writer’s visits did not call for careful examina- tion of them. Mathematical models do not ap- pear to be in much favor. A large, cheerful room, called the “ Confe- renzzimmer,’ is always set apart for the social and official use of the faculty. Along fhe tacuity- the side walls are ranged desks and ior. lockers for each teacher, and through the centre extend long tables amply provided with writ: ing material, including red ink. 40 Teaching of Mathematics in Prussia This room is the working centre of the insti- tution. From here pulsates the life-current of | the work, and the influence of this simple, easily provided centre gives tone, vigor, and unity to the activities of the entire teaching body. Here each teacher meets his colleagues several times daily, and exchanges a few social phrases with them or arranges some detail of their common work; here each keeps his needed books and papers, refreshes himself, rests quite free from responsibility during the ten-minute pauses and works in quiet when he has a free hour; and here finally, the Director drops in frequently for a little chat or to speak about this, that, or the otherof the routineoftheday. Inall these ways the constant use of this room must contribute in no small degree to unity and efficiency of work. The sessions are held on all six week-days and the arrangement of hours in the day is fixed by each school in accordance with localneeds. The vacations vary slightly, but usually are about twelve weeks in length, divided as follows: Sessions. BLAStOT. i 2st. 24 iol os tle aR eee 16 days Woihitsuntigey 0 0 ae be le eee Wikee Summer (usually including July) 38 “ Michaelmas. (2 i ig te as one Oct ACO ISOS: p's \e's)e wtp hie ek et ea 1G ies Che Institutions AI In addition there are various single holidays, and in summer the session is often omitted or abridged on hot days. The year is divided into two semesters, the Michaelmas and the Easter; if the number of pupils is large enough, there are two full sets of classes, in one of which the year begins at Michaelmas, in the other at Easter. When this is the case, a pupil who fails to be promoted falls back only six months. When there is only one set of classes the year usually begins at Easter, though not invariably. An interval of ten minutes intervenes between the ringing of the bell which terminates one exercise and the ringing which is the signal for the beginning of the next. During each of these pauses the pupils are re- quired to leave the room and take exercise in the yard; if the weather is bad they remain in the corridors. A pupil remains in charge of the books and clothing left in the classroom, and in each corridor and in the yard a teacher exercises supervision, but otherwise all are free during the pause. It is a period of relaxation and refreshment for all. Both pupils and teachers bring a supply of sandwiches to be eaten inthe pauses. Noises are tolerated freely both in the corridors and on the playground; the pupils run about and shout, but I saw no concerted plays. Indeed, the American visitor The pause. 42 Teaching of Mathematics in Prussia to Germany is soon struck by the fact that the children gambol about aimlessly and do not play games. As a few minutes are necessary for the pupils to return to their rooms after the bell has rung, the actual period of instruction of each “hour”’ is about forty-seven minutes. VII The Curricula The first plan of studies uniformly followed throughout Prussia was issued in 1837. Prior to that only the rules for final exam- First inations (Adzturzentenexamen), first is- curriculum. sued in 1812, and isolated ministerial orders, were binding upon the schools, which on the whole arranged their curricula independently. The class system was introduced throughout Prussia in 1820, and there have been during the present century four revisions of the course of instruction—v7zz., in 1837, 1856, 1882, and 1892. The last revision was preceded in the autumn of 1890 by a Conference called by the minister of education to discuss a number of theconfer- questions submitted by him. The ence of 1890. forty-four members included representatives not only of the schools concerned, but also of the Universities, the Educational Administra- tion, the Church (both State Church and Roman Catholic), and the Army. Among the members who are well known in America were Paulsen, Helmholtz, Virchow, and Zeller. The delibera- 43 44 Teaching of Mathematics in Prussia tions of the conference extended over several weeks and were opened and closed with an ad- dress by the Emperor in person. The com- plete stenographic report of all the delibera- tions has since been published’ and will repay careful study. Among the important problems considered were the question of obligatory Greek (the spe- cific question at issue being further preroga- tives for the real-institutions), the extent of the Latin instruction, and the overburdening of the pupils. The first resulted in no additional preroga- tives for the real-institutions; the second, ina marked diminution of the number of hours of instruction devoted to Latin, an increase in the number of hours given to German, and the abolition of the Latin theme as an end in itself (Zielleistung); and the third resulted in a dimi- nution of the hours of instruction in all the schools (see curricula below), and in a sharp demand that home-work should play a very minor part (see methods, etc., below). It is with much regret that this mention of these instructive deliberations is permitted to suffice. The interested reader will find the published report very suggestive. Passing to the consideration of the curricula 1 Verhandlungen tiber Fragen des hiheren Unterrichts, Berlin, 1891, pp. 800. The Curricula 45 themselves, we give first the detailed curricu- lum of the gymnasium as fixed by the plans of 1892. The abbreviations already fhe currice mentioned (p. 34) are used for the «lum of the classes, and the figures denote hours #77" of instruction per week throughout the school year. CURRICULUM OF THE GYMNASIUM. VE Ve v. IIIB. IIIA, |ITB. ITA.} IB. } IA.| Total. Christian Religion...... ETE We ee 2 2 2 heads | 19 ermancncdeiene. ss sass2 4 } x t 3 2 2 3 Suwa are (DEN a als Gee ee ee 8 8 7 7 7 6 6) (76) 62 GG APL ltt Psee 6 6 6 Gore Go 36 RONG Heerassydie'cs sys clasts sie 'e Satta ted 3 3 2 ZalesttenLo History and Geography.| 2 | 2 | 4 3 3 3 Shes: (i 20 Mathematics): 6 3..:.. AM irae Wid. 3 3 4 Fide ehh icy Natural History........ 2 le) to 2 a ed SW aid pone 8 Physics, Chemistry, and Mineralogy......... By HAE 2 2 EAP ie Bl lee fe: WISE Kero cle itecls ceases Deo el | ae ae 4 4 PAWS is cde esse dues ce zaS 2 2 7 8 aL otalin =). pe teapy « 25) 25 oorlego) 44,30 28 | 28 | 28 | 252 NorESs. xz. German and Latin in VI. and V. are to be taught by the same person if possible. 2. Three hours per week of physical culture are required of all classes. 3. The portion of the table to the left of the heavy line constitutes the curric- ulum of the progymnasium. 4. The instruction in the Christian Religion is given according to the tenets of the State Church (Protestant) or those of the Roman Catholic Church, or both, as circumstances may demand. Jewish pupils and others who cannot conscientiously attend either the Protestant or the Roman Catholic instruction are dispensed from the requirement in the Christian Religion, on satisfying the authorities that they are receiving an equivalent amount of instruction in tenets of their own religion. If the number of Jewish pupils is sufficient, instruction in the tenets of the Jewish religion is given in the school, in the amount re- quired by the curriculum, 46 Teaching of Matbematics in Prussia The above plan indicates sufficiently clearly the way in which the hours in each subject are Synopsis of 1Stributed among the various classes. other In the other kinds of institutions and curricula. in the earlier curricula the distribu- tion is, in principle, the same, so that it will suffice to give for them the totals correspond- ing to those contained in the last column of the above table. (The unit is one hour of instruc-. tion each week throughout one year.) CURRICULA OF THE PRESENT CENTURY. Real- Oberreal- Gymnasium. gymnasium. schule. 1837|1856| 1882] 1892|| 1859} 1882/1892) |1882]1892 Christian Religion. ; oi) -uiespas0s 18 | 20 | 19 | 19 |; 20 | 19 | 19 |) 19 | 19 Gernian ices esic ema le cele opie sere 22 | 20 | 21 | 26 || 29 | 27 | 28 || 30 | 34 GeV EYIa bere rive: slats Bis oe's inte aoe tations 86.|'86 | 77.}'62 || 44) S45) 4st ei Greek Oo aasclo can motes ww atelepnarrs 42) 42e) Aowls36.Wl 2.) cee joa giete PROP T ag techs cre aves eve sie ferale, Clete oleae 12, | 17 | 2x] 19 || 34 | 34 | Sr uh) SOllay IBPISh. Geettciie ae eat wnat oars ae fae e pose Plate afl 20°] $20 Te Siete History and Geography......... 24 | 25 | 28 | 26 || 30 | 30 | 28 || 30 | 28 Mathematics icity slesisis as sites oe 32 | 32 | 34 | 34 |! 47 | 44 | 421) 49 Woa7 Natural History 6| &| 20] 8 || 18 | x2 | 22 || 13 | 22 Physics fo ead a ieiata eg i 6} 8) z0*}) 8. | a2) een erie Chemistry 220 spic aise voto siale'sic's nie eat deel Aakoat rss 81 206 ap 9 | 11 Writing lees seein cemeteries af Grane: 4 7 4 4 6 6 LAWINH, |.) dalenis es pik ahaielatele's ie eats 6}° 61 6} S j)20) s8ahrenh reneieae eLotal aovcssiectaec oe coetlen 258 | 268 | 268 | 252 || 285 | 280 | 259 || 276] 258 * Includes the elements of chemistry and mineralogy. The ministerial rescript offers the following regulations concerning the instruction in math- ematics in the gymnasium: The Curricula 47 a. General aim of the instruction.—Facility of calculation with numerical quantities, and their application to the usual circumstances 4) cuericue of everyday life. Literal arithmetic to tum in math- the binomial theorem for positive in- = “4° tegral exponents and algebra to quadratics, both inclusive. Plane and solid geometry. Plane trigonometry. The idea of codrdinate and some of the fundamental properties of conic sections. In all of these subjects not simply an intelligent knowledge of the theorems is to be reached, but also skill and facility in their application. 6. Scope of instruction.—The topics treated in the various classes are as follows: VI. Four hours per week.__Review of the fun- damental operations with whole numbers, both abstract and denominate. German weights, measures, and coins, with exercises in the deci- mal notation and the simplest calculations with decimals. V. Four hours per week.—Divisibility of num- bers. Common fractions. Simple exercises in proportion, to be solved by reduction to unity. German weights, measures, and coins as in VI. IV. Four hours per week— Arithmetic, two hours. Decimals, simple and compound pro- portions with integers and fractions (exercises from practical life). Plane geometry, two hours. The straight line, angles, triangles. 48 Teaching of Mathematics in Prussia IIIB. Three hours per week.—Literal arith- metic, one hour. The fundamental operations with absolute numbers, restricted to the most necessary matter. In the exercises equations of the first degree are also to be used. Plane geometry,twohours. Parallelogram. First part of circle. | TILA. Three hours per week.—Algebra (first half-year, one hour; second half-year, two hours). Equations of the first degree with one and several unknowns, herewith exercises in fractions. Powers with positive integral ex- ponents. The most necessary things concern- ing radicals. lane geometry (first half-year, two hours; second half-year, one hour). Theo- rems concerning quality of areas of figures. Computation of the areas of rectilinear figures. Beginning of the theory of similarity. I1B. Four hours per week.—Equations, includ- ing simple quadratics with one unknown. Powers with negative and fractional exponents. Concept of logarithms. Exercises in compu- tations with logarithms (five place). Compu- tation of circumference and area of circle. Definition of trigonometric functions. Trigo- nometric computation of right and isosceles triangles. The simple bodies, with computa- tion of lengths of edges, surfaces, and volumes. IIA. Four hours per week—The theory of powers, roots, and logarithms, Equations, in- The Curricula 49 cluding quadratics with several unknowns. Arithmetical and geometric progressions. Conclusion of the theory of similarity, golden section, something on harmonic points and pencils. Plane trigonometry, with exercises in the computation of triangles, quadrilaterals, and regular polygons. IB. Four hours per week.—Review (by means of exercises) of the algebraic work of the earlier classes. Compound interest, annuities, imagi- nary quantities. Completion of trigonometry (addition theorem). Solid geometry and mathe- matical geography on the sphere. IA. Four hours per week.—Binomial theorem for positive integral exponents. Conclusion of solid geometry. The notion of co-ordinates and some fundamental properties of conic sec- tions. The degree of thoroughness with which these various topics are handled may be judged from the texts used, the actual work seen in the class- room,and the examinations set at the close of the course. The specimens of the examination papers which we shall give later will perhaps enable the reader to form his own opinion as to the scope of the instruction. The text-books and the class instruction will also be discussed in another connection; it may be said here that all the available information seems to indicate that on the whole the German work is at least 50 Teaching of Mathematics in Prussia equal in extent and thoroughness to that of thé better American schools. c. Methodic remarks.—The ministerial rescript adds a few directions under the title “ Method- ic remarks,” among which are the following: The teaching of arithmetic is to aim at secur- ity and facility in operations with numbers. That it may be in harmony with the following algebraic instruction and prepare for it, the re- views of the fundamental operations in Sexta, as well as the treatment of fractions in Quinta and Quarta, must be based upon mathematical form, and the handling of parentheses must likewise be continually practised. In fractions the pupil is to be taught to operate with frac- tional parts as concrete things. The instruction _in arithmetic as such stops in Quarta, but se- curity in computation is to be maintained by continued numerical exercises in the algebraic instruction of the following classes. Strict adherence to the work assigned to each year is an absolute requirement. As it is more difficult in mathematics than in other subjects to replace deficiencies in elementary attain- ments by private industry, and as experience has shown that the difficulty which this subject sometimes presents to the pupil in the upper classes is almost without exception due to deficiencies in the foundations, conscientious strictness in the promotion of pupils becomes The Curricula sr the more an urgent duty toward the pupils themselves. We pass to the mathematical curriculum of the Realgymnasium and of the Oberrealschule. The scope of the instruction in the Mathematics real gymnasium is as follows: is hii Algebra, including the proof of the schools. binomial theorem for arbitrary exponents and the solution of equations of the third degree. Plane geometry, including the theory of har- monic points and pencils, and points and axes of symmetry. Solid geometry and the funda- mental propositions of descriptive geometry. Plane and spherical trigonometry. Introduc- tion to the theory of maxima and minima. Plane analytic geometry. In addition to these, there are required in the Oberrealschule : The most important series of algebraic an- alysis. Equations of the fourth degree and the . approximate numerical solution of algebraic and transcendental equations may be taken up at the option of the instructor. In all these topics the work is to give prac- tice in the application of the theorems, as well as to lead to a mastery of the proofs themselves. The directions for the work in mathematics have been very sharply criticised by the math- ematicians in the schools affected. It is not necessary to reproduce these criticisms here or 52 Teaching of Mathematics in Prussia to make comments of our own with the one exception of calling attention to the sad defi- Fie ciency in the “General Aim of the strictures. Tnstruction” which is set up to guide the teacher in his work. No unifying principle is offered for making the mathematical work a harmonious whole, no suggestion of treating this subject as a portion of general culture; nothing but the attainment of a speci- fied amount of mathematical technique is fixed as the aim of the work. The distribution among the various classes Distribution of Of the hours allotted to mathematics the hours. in the different curricula, appears in the following table: Gymnasium. VI. Vv. IV. IIIB. IIA. IIB, IIA. IB. IA. Total. 1837....0. 4.4 (3 3. 3. 34. 4. ee 1856...... 4°93 3 3 BS 4 SS 1882... .06 AR BOB 3 4 A eee TROIS suid 4.4 (4 63 2 3° ©4014 % > Sage Realgymnasium. ISSO. 24 6:0 5 4 6 6 6 5 5 5 5 47 1882.....- §S «4.5 5° 96 ).55.9 95) 5 345 eee 18QI.eeeee 4 4 4 .§) $ °45.. 45 35 Spee Oberrealschule. TESS 55 ya 8 6 6 6 6 5 5 5 5 49 I8OL. 250. 5 5 0726 5 5 5 5 were}: VIII The Tnstruction in Matbematics The arrangement of the classroom, the benches and the small blackboards, have already been mentioned. Wet sponges are ctassroom used as erasers. Light (a/ways from cUatiasies the left of the pupils), heat, and ventilation were as arule adequate. The lower classes which I saw had from thirty to thirty-five pupils, while in the three upper classes the number ran from nine to twenty (the last number was exceeded in only two instances, Obersecunda with thirty pupils). The pupilsare all assembled when the teacher comes and when he enters they invari- ably rise and remain standing until he bids them to be seated. They assume the same attitude when he leaves the room at the close of the hour, and also whenever any other member of the faculty, or a visitor like myself, enters or leaves the room. The pupils are addressed as Du to the close of Obertertia, thereafter as Sze. The first thing which impressed me in the class- work, and that which remains finally the most prominent characteristic, was that the ¢eacher 53 54 Teaching of Mathematics in Prussia teaches. We does not “hear recitations;” he does not examine the pupils to see whether or Theteacher nOt they have learned some assigned poouetce: matter from a book; this custom seems happily quite a thing of the past here. At times he imparts new knowledge himself, es- pecially by way of definition and introductory work, but most frequently he leads the pupils on by skilful questions themselves to discover new truths. In the development of new propo- sitions the teacher guides the work, but the pupils suggest step by step what is to be done next. Home-work and the study of books are very minor features; by far the heaviest stress Stresson 15 laid on the class-exercise. Here; theclass- under the carefully planned instruc- exercise. = tion, under the direct influence of the personality of the teacher, the progress is to be made. Private work and the study of text-books have simply the purpose of fixing in mind or giving practice in that which is sup- posed already to have been learned. The teacher is the source of the pupil’s knowledge and the authority on which he builds. “ What does the book say?” is a question never raised in a German school; in all my visits, I heard no books referred to, except collections of exercises. Whether it is best to train pupils to such Tbe Mnstruction in Mathematics 55 marked dependence on personal, oral guidance in the acquisition of knowledge is open to ques- tion. I am here and throughout simply stat- ing the facts as I found them without discussing their merits either when regarded alone or in comparison with other systems. An educa- tional system is far better judged by the results it accomplishes under fair and sufficient trial than by any quantity of theorizing about it. The results reached under the Prussian system will be discussed later. This is, however, the proper place to say that the Prussian system does produce most excel- lentteachers. Thethorough preparation which they are required to undergo both in the scien- tific and the practical pedagogic lines tells dis- tinctly in their work ofinstruction. There were many remarkably strong teachers among those whose classes I visited, and among them all there was only one that could be called un- questionably poor. If I were to describe the method of instruc- tion taken as a whole by a single phrase, I should say it is the “ Socratic ee method,” the method of skilful ques- Socratic tioning, of leading the class on tothe ™"*" desired goal by a series of questions, each usually fairly easy to answer in itself. Except in case of review-questions to refresh the mem- ory or to recall the material needed for the day’s 56 Teaching of Mathematics in Prussia work, the questions have a clear didactic pur- pose and value, and generally give evidence of having been carefully planned. Every bit of the hour’s work is vitalized by the teacher; there is nota minute when his voice is not heard and there is also not a minute when his voice only is heard. I was especially impressed with the general custom of dividing the work into very simple steps, and of repeating each new fact established over and over until it seemed that it must be imbedded in the mind of the slowest, before going on to the proof of the next. This is very distinctly characteristic of the German class- room. The questions are very simple, often half suggesting the answer, but still leaving some- thing for the pupil to think out and add. One director, in praising his leading teacher of mathematics most highly, said: “He hammers away and simply makes the pupils follow the work; the secret is that he works for the slowest.” It may seem that this mode of procedure is to the disadvantage of the brighter pupils by holding them back to the pace set by the weaker portion of the class. Indeed, several German mathematicians have remarked to me upon the slowness of the progress and have recalled how irksome. this had sometimes proved to them as pupils. The problem of how to care for the The Mnstruction in Mathematics 57 weak and the average pupil without holding back the talented pupil to his detriment seems to be as far from solution in Prussia as in America. If, however, the galaxy of mathema- ticians who have sprung from the benches of the German gymnasia be taken into considera- tion, the question may well be raised whether or not the retardation of the gifted pupils is in fact to their detriment. The answers are always given in complete sentences, and clear and distinct enunciation is insisted upon. Every lesson in math- ematics is thus more or less of a lesson in German. Considerable stress is also laid upon the oral solution of easy exercises. In review, quite complicated problems are thus proved. I heard, for example, boys about thirteen years old (Untertertianer) prove the Pythagorean theorem with no figure whatever before their eyes. Different pupils did not use exactly the same lettering for the figure. The teacher informed me that this was not a mere feat of memory, but that the pupils could fol- low the proof on an imagined figure and that they enjoyed this kind of work. I saw this in various forms in several classes, and the pupils enter into it with considerable zest. Simpler new theorems are also proved orally, in some cases with the figure on the board; in others the figure was constructed and all were allowed Oral work, 58 . Teaching of Matbematics in Prussia to take a good look at it and fix the image in mind, when the figure was erased before the proof was begun. Work in concert is effected in the Diarium or exercise-book. Some exercise is dictated The exercise. Dy the teacher; the pupils work it book. simultaneously, one reading as he works; the same pupil reads only one or two steps, so that quite a number are called on before the exercise is finished. The reply which the pupil makes seems to be the only means which the teacher has of determining in how far each pupil has worked along with the others and understood the step taken. The exercise-books are not usually inspected either during or after the class-work. Sometimes the teacher or a pupil works on the board, the others working along on paper, or looking on and dictating in response to the questions of the teacher ; some- times the teacher works on paper with the class; sometimes he has the exercise in hand, already worked out. Sometimes the result found is discussed, reformulated by members of a class until a satisfactory form is reached, and then copied as a theorem for future use. The “Diarium method” could readily be adapted to work at the board by the whole class in concert. Whether working in the diarium or at the blackboard, the pupils are trained from the > f The Wnstruction tn Matbematics 59 very beginning to read aloud distinctly what they write as they write 7t—to “ chalk and talk.” This habit might well be cultivated «cham ana in American pupils. One of the talks chief advantages of an oral explanation is that one sees the proof growing and taking shape, and comments can be made on each step as itis taken and its correctness and purpose satisfac- torily explained before the next step is made. In the same way, figures may be constructed line by line as required. Indeed, by far the best way to master a proof which is put before one in complete form is to write out the proof step by step on paper, constructing what fig- ures may be needed as the proof proceeds. If complete figures are prepared beforehand at the board, and the reasoning is written out in full before any oral explanation or discussion takes place, the possibilities of oral work are very imperfectly utilized. In an oral explana- tion the word of mouth is of chief moment; the writing is auxiliary and of the character of a record of what has been explained. The attainment of any degree of skill in the threefold activity of performing an operation, describing it orally, and recording it in symbols, requires systematic and persistent practice throughout the entire work in mathematics. The German boys in the lowest classes were as ready in writing and reading simultaneously 60 Teaching of Mathematics in Prussta the operations they had to perform as were the boys in the higher classes in their more complicated work. This is so because the teachers insist on it from the very beginning, by example as well as precept. It might be well to encourage the pupils to practice at home explaining proofs aloud and writing them out simultaneously. In addition to its direct purpose, this practice would also contribute to fixing the spirit and methods of the proofs themselves more firmly in mind. It should therefore be confined strictly to proofs which have been fully explained in class and clearly understood, lest, otherwise, errors and erroneous conceptions should become more firmly rooted. The following lessons may illustrate some of the characteristics of the instruction that have Alessonin JuSt been described. The first, a les. - aigchr, son in algebra, is taken from my notes of a class visit, while the second, a lesson in geometry, is a model for the study of begin. ners, set up in a German work on the teaching of mathematics. The algebra lesson was given in Untertertia, the first year in which algebra is taught, the minimum age of the boys on entrance into this class being eleven years. 1Reidt, Anleitung zum mathematischen Unterricht an hoheren Schulen, 1886, p. 31 et seq. The WTnstruction in Mathematics 61 First, the following problem from the book of exercises in algebra! used in this class was taken up. ee 2 tee Neng)! hy 608 Ta 6 st be All wrote the problem, one (John, say) read- ing aloud as he wrote and adding: “ We seek first the common denominator.” Teacher. “Yow do we do that? By a rule?” John. “No, by inspection.” Teacher. ‘“ Right. What is the common de- nominator ?”’ John. ‘“ Twenty-four.” Teacher. “Right. What do we do next, Henry?” Henry. “We multiply both members by twenty-four.” Teacher. ‘“ What is the result, William ?” William reads as all write, “t2r— 8x + 64 —4r + 34+ 24 = 264.” Teacher. “What do we do next, Karl?” Karl. “We unite the terms in the left mem- ber.” 1Bardey, Methodisch geordnete Aufgabensammlung, mehr als 6,000 Aufgaben enthaltend, 23te Auflage, 1897. 62 Teaching of Mathematics in Prussta Teacher. ‘Give the result, Fritz.” All write as Fritz reads and writes, Pate 20Ane Teacher. ‘ What do we do next, Peter?” Peter. “We divide both sides by eleven.” Teacher. ‘What is the result?” Peter reads and all write, AE met Fb A problem just like this was worked simi- larly, and then, as this was one of the days on which home-work in mathematics is to be as- signed to this class, three problems of precisely the same nature were assigned by page and number from the book of exercises for home- work, viz.-} x a = 2th BE cig =, ashen 1 2he — 3444+ 544-344 +1=-4%. ae ag 2 es 2b eet ea. This constituted the entire assignment for home-work. 1 Bardey, Aufgabensammlung, p. 101, Nos. 73, 74, 75. The Mnstruction in Matbematics 63 Next, the following problem was taken up, all writing and one reading as usual. 2(7x — 10) — (50 — x) = 20. Teacher. ‘ What doesn’t please us here?” Various pupils raise hands and reply as called on by the teacher. “ The parentheses.” “ The known numbers on the left.” “ The fractions.” Teacher. ‘“ Which shall we remove first?” The pupils express different opinions. The teacher points out that the most practical order must be determined in each problem—“a mat- ter of feeling ”’—and then indicated that in this problem it would be easiest first to remove the fractions, then the parentheses, and then to rearrange and solve. All of this was carried through step by step as in the previous case. Then oral work was taken up. First, the expansions of (a + b)* and (a — b)* were re- hearsed both as formule and in words, and then a number of exercises were given, the teacher writing on the board and the pupils reading the results as called upon to doso. Very easy exercises were given at the beginning while those at the close were of the difficulty of the following : (44 — ay); (52?2+4)*; (+0; (4 — #4 — 2). 64 Teaching of Mathematics in Prussia Then the formulz for (a + b)* and (a — b)8 were deduced and repeated a number of times, and the hour came to a close. The geometry lesson is supposed to be given to the class Quarta, in the first year’s Alessonin Study of geometry, the minimum geometry. ace for admission to this class being eleven years. “The teacher draws a triangle ABC upon the board, and then questions the pupils some- what as follows, the pupils being called on singly by name in as lively alternation as pos- sible : How many angles has a triangle ? Name an angle of the triangle ABC. A sec- ond. A third. The teacher draws and defines an exterior angle, CAD. Who can draw another exterior angle? (Done repeatedly by various pupils.) How many exterior angles can be drawn at one vertex of the triangle? How many exterior angles can be drawn al- together? What are two exterior angles at the same vertex called with regard to each other? What therefore do we know as to the magni- tude of these two angles? How many exterior angles differing in size can a triangle have at most? The Mnstruction in Mathematics 65 Why is it customary to speak of only one exterior angle at each vertex of a triangle ? In view of this custom, how many exterior angles would a triangle be said to have? For convenience, the letter at any vertex shall be used to denote the interior angle and the letter primed the exterior angle at that vertex. The notion of adjacent angles is supposed to have been explained earlier in the course. What are an interior angle of a triangle and its adjacent exterior angle called with respect to each other ? What theorem holds for two such angles? We wish now to compare the magnitude of an exterior angle with that of the two non-ad- jacent interior angles. For this purpose, we regard AB and AC as two non-parallel straight lines cut by a third, BA; the last produced forms the angle CAD or A’. In what position are A’ and B with respect to each other? Likewise A’ and C? Can therefore A’ = B, or A’ = C? To compare the magnitudes of the angles we draw AE parallel to BC and divide A’ into the angles CAE and EAD, which angles we call ~« and y respectively. Since AE is parallel to BC, there is another angle in the figure equal to y; what is it? Why are y and B equal? Which are the parallels and which is the secant line ? 66 Teaching of Mathematics in Prussia The teacher marks the equal angles with the same mark. Is there also another angle in the figure equal tox? What is it? The teacher also marks these equal angles with the same mark, different from that used with the previous pair. Why are these two angles equal? Whichtwo lines are now the parallels? Which the cutting line? Since y = B, and x= C, how large is x+y? To what is therefore the angle A’ equal? What theorem have we thus found ? The proof is now repeated synthetically, first with the same figure, then with a different exterior angle of the same triangle, or also with an entirely different triangle, something as follows: The teacher produces BC beyond C and asks: What do we assert concerning the exterior angle at C? What auxiliary line shall we draw to facilitate the proof? What pairs of angles are now equal? Why? (To avoid breaking the main course of thought, the parallels and the secant line are not now called for in detail.) What follows from the equality ? Karl, give the entire proof once more. (To hold the attention of the other pupils the teacher The Wnstruction tn Mathematics 67 interrupts, if necessary, and calls upon others to give the reasons for statements made by Karl.) | In the next hour one or more repetitions of the proof are given in the same way until, if possible, all the pupils are able to present the proof in a connected manner.” “Many a beginner in teaching will regard a large part of the questions in the above speci- men as very superfluous and as use- _ pjections lessly squandering time, because he considered. thinks it may be taken for granted that the pupils know the answers. He will even fear that such seemingly trivial questions will not only not attract the pupil, but actually bore him and cripple his interest in the work through default of rapid progress to new matter. But if he makes the experiment of teaching in the manner - indicated, he will soon become convinced of his error. He will see that the great majority of the pupils are eager to participate in the dis- cussion, that they compete for the privilege .of answering the questions asked, and are rejoiced to know and to do something. He will also notice that the answer to these questions are by no means evident to some pupils, and he will be obliged to take special pains with these pu- pils. He will often enough be amazed at the colossal stupidity of some of the answers to his questions, but he will also be gratified to see 68 Teaching of Mathematics in Prussia how, gradually, stupidity gives place to com- prehension, and the progress of the pupil, at first so slow, becomes more and more rapid, in consequence of the pupil’s own mental exer- tions. It will not be long before the stronger pupils at least will be able to make simple proofs without the previous assistance of the teacher.” The manner of the teachers was usually of a military sharpness, though not unkind; in many Manner of the CaSe€S it was mild, in a few genzal, and teachers. only exceptionally unkind or irri- table. The routine directions, especially, are given with the snap and precision of a military command and are met with an obedience equally military in its promptness and unanimity. This custom, together with the fact that the time spent by a teacher with the same class is meas- ured in years, permits the development of an in- formal code of directions (words and even gest- ures), by which the time occupied in giving and executing routine directions may be con- siderably reduced. There was often a sharp rattling fire of questions to which the an- swers came with corresponding promptness and precision, but at crucial points, where it seemed necessary for the pupil to collect his thoughts, ample time was allowed. The pupils were not only not hurried, but were openly encouraged to take time to think. Tbe anstruction in Mathematics 69 “Take five minutes if you wish, oz/y make no mistake.”’ In commenting upon the work of pupils both the warmth of the praise for the good work and the severity of the censure for the poor work were more intense than they would have been under the same circumstances in America. It is safe to say that, despite the fact that many of the teachers of mathematics in our secondary schools are women, the emotional treatment of the instruction in Prussia is decidedly more de- monstrative than here. Concerning the written exercises nothing need be noted except the neatness of the papers. Isawanumberofsets,both ioe of final examinations and of class- of written work, and was present a few times it when the latter were returned with criticisms. All are written on uniform paper and any lack of neatness and mechanical accuracy is sharply criticised when the papers are returned. The result is that even the papers of the final exam- ination, when perhaps hurry and flurry might palliate careless writing, are to my American -eyes models of neatness. This standard can be reached only by unremitting insistence upon neatness from the very beginning and through- out the entire nine years. Indeed, I was told that the few papers which I singled out in one case as not up to the standard of the others, 70 Teaching of Mathematics in Prussia were written by pupils who had entered the Gymnasium late and thus had not had the same drill as the others. The schedule of instruction is arranged so that any one class passes through the hands of as few different instructors in each subject as is practicable. During the nine years of the course of study the pupil has only one or two instructors in mathematics in the smaller institutions and from two to four in the larger institutions. Even in the latter, the aim is, if possible, to arrange the schedule of instruction so that the work in mathematics of the last three years shall be under one instructor, in order that the pupils may be carefully and sys- tematically prepared for the final examination. Whatever the number of teachers, the entire mathematical education of the boy from the Homoge. Clements of arithmetic to those of neity of analytic geometry takes place in one instruction. institution under one management, guided by the close supervision of the same director and under the tuition of men of the same scientific training, who are colleagues working in close contact, with opportunities for intimate interchange of ideas. Besides the director, the senior professor of mathematics gives more or less attention to the work of the younger instructors and thus contributes to uniformity of tone and spirit. The Wnstruction in Mathematics 71 On the desk of each class there lies a large book, very durably bound, devoted to the rec- ord of the class’s work. One page is he ctass- allotted to each day’s work and one pete book serves a year. The page is provided with four columns, respectively for absentees, remarks, record of matter treated in the hour, and assignments. Horizontally, the page is ruled for each school-hour, so that at the close of the day a complete record of the day’s work in the class appears on the page. The home- work assigned being recorded, each instructor can see what the class is already required to prepare, and govern his own assignment ac- cordingly. Usually, the days on which home- work may be required at all are fixed for each subject, and I have at times seen this schedule posted on the wall of the classroom together with the hour-schedule for the class which is invariably there. The class-book is filled out beforehand as far as possible (dates, hours, sub- jects) by the Primus or first boy of the class. The other entries are made and signed by the teacher immediately at the close of the hour. In the column for remarks, anything not in the usual routine is entered, such as pupils excused, pupils misbehaving, reprimanded, punished, or doing very bad work. A few specimens will best show the nature of these entries. They are taken from some 72 Teaching of Mathematics in Prussia class-books which I was kindly allowed to look through at leisure. They are all taken from Specimen ifferent dates during one year, and entries. were made in one institution by va- rious teachers in different classes. A. was excused at 9.55. (The hour closed at ten.) B. occupied himself with outside matters dur- ing the hour. C. is not prepared. D. does not know the Homer verses by heart (the second time). D. does not know the Homer verses again (the third time). E. was reprimanded because of disturbance during the pause. F. is to blame for repeated lack of industry in Homer. The class was very noisy during the pause. G. knows nothing, and besides that answers pertly. H. is punished with one hour “ Karzer”’ (im- prisonment) because of persistent lying. J., on account of repeated unbecoming con- duct, receives the order to clean the sponge thoroughly daily. K. speaks with his neighbor during the written test and is punished with one hour “ Karger.’ The class-book is under the constant ispec- ae oe The fnstruction tn Mathematics 73 tion of the ordinary, who thus keeps informed as to the work of the class, and the director is required to inspect all the class-books at least once each week. The pupils are also at liberty to examine it. All home-work is regarded as supplementing the work in the class and not asan integral part of the course. Its purpose is either the cultivation of neatness and order- liness in making clean copies of class-work, the memorizing of indispensable material, the fixing of what has already been learned, or the training to independent activity. Matter that has not been thoroughly explained in the class, so that the class as a whole understands it clearly, is never assigned to be studied privately by the unaided pupils. The quantity of the home-work is to be kept as small as possible. As maxima for the various classes, the following are officially suggested: | Stlmeveoely, LLIB, “LITA IIB: IIA; IB. TA. Meek 2 a 24 Sat 8:3 3 3 hours daily. Home-work. As already mentioned, the distribution of this time among the different instructors is arranged beforehand by the director or by the instruc- tors interested in conference, fixing the days on which each instructor may assign any home- work and the maximum of time which he is then at liberty torequire. The permanent record of 74 Teaching of Mathematics in Prussia every assignment in the class-book enables col- leagues and superiors to keep track of each in- structor and see that he does not exceed his al- lotment. I learned the assignments in a few cases, and it seems, roughly speaking, that the amount Specimen Of home-work in mathematics re- allotment. quired per week in Prussia is consi- derably less than twice the amount required per day in the United States. As specimen of the allotment of time for private work, the follow- ing table may serve, giving the hours per week, as fixed for each subject and class in the Kgl. Realgymnasium, Berlin. VI. Vv. IV. | IIIB.) IITA.) LIB, | TLAS) SIBSeEAS Christian Religion| 14 1} 14 13 1} It 1} 1} 14 German. cocnnens 1 I I I I 1¢ 1; 2 2 Latino nearee 3 3 4 2} 2t 2+ 2 2 2 Krenchjcysc pb cntice aoe. Uh eee I 13 2 2 2 2 2 Ron gliahh s+ sie:s ay’ t | 5 vival Same ae toe ees I 1} 1} 2 2 2 Historyacesaes ce ° ° 4 4 4 I 1¢ 1¢ 2 Geography ...... 4 3 3 & 3 4 | vise eee Mathematics, ... I I 14 2 2 2 24 2% 24 Natural History. + 3 4 b 4 Men ee ee Physios Ske eal oe ee ome cae oP aie natalie ee lle or a I 1t 1 Chemsintry 6) sik So hina tin ae oe pavtne OnE PRE Salah eile bree £ I I Total per week..| 7} 74 | roy | r1f | 12$ | 134 14 | 16 16$ Total per day....} 14 1} 18 133 | 23 2} 23} | 2§ 25 Maximum per day suggested by Ministry...} 1 I 2 2 2¢ 2t 3 3 3 In noticing the allotment to mathematics, it should not be forgotten that the scope of the work and the number of hours of instruction Tbe Unstruction in Mathematics 75 in mathematics in the Realgymnasium are considerably in excess of what is done in the Gymnasium. The weekly amount is divided up into one, two, or three daily portions for each subject, and the days on which these allotments may be used are specified in such a manner that the aggregate daily assignment for private work is not far from the average on any one day. There are no “study hours” for pupils at school. While there, their time is occupied entirely with class-work. Ministerial rescripts of December 24, 1833, and August 16, 1860, require that some text-book be used in mathe- matics. The decision as to which book or books Text-books. 1 The following is a free translation of these rescripts. (1833) ‘‘ The Ministry has had occasion to note that in many Gymnasia the instruction jn mathematics is conducted without a definite text in the hands of the pupils. In mathematics, if any- where, a brief text, suited to the needs of each class, is indispensa- ble. . . . In order to counteract these and other evils which have hitherto been more or less marked in the mathematical in- struction in the Gymnasia by reason of the lack of a definite text- book, the Ministry wishes herewith to fix that from Easter of next year a definite text-book is to be used in the various classes of all Gymunasia, that this text-book is to be in the hands of the pupils, and that no further attention is to be paid to any objections to this regulation which the teachers may raise. (1860) ‘‘The large number of text-books in mathematics and physics which are in use is an evil of considerable import. It is therefore very desirable that the use of those which have not proved 76 Teaching of Mathematics in Prussia are to be used is made by the local authori- ties, but their choice is limited to such works as have been approved as sufficient for the purpose by the Ministry or the Provincial Board. Many books are used in one institution only. The presumption might be that in these cases the authors are teachers in the institu- tions using the book, but this is by no means always the case. To counteract the tendency to multiply text-books the authorities are be- coming more slow in placing new books on the approved list. It is now required that institu- tions with which the author is not connected shall have expressed their desire and purpose to use the book as text before it is placed on the approved list. Still, with all this discour- agement, the German teachers are writing new text-books in mathematics by the score every year. The text-book adopted is in many cases re- garded as named rather to comply formally with the regulation than for the purpose of actually using the book. This statement is strong books be still further discontinued and that they be replaced by more suitable works; nevertheless, the educational administra- tion will refrain now, as heretofore, from any direct constraint in this matter. The books adopted are often used too little, still, in attempting to enforce more use of the books, the danger would be incurred of hampering the more important effectiveness of the free individuality of the teacher.” The Tnstruction in Mathematics fir based both on information which I obtained from German teachers and on my own obser- vation. In all my visits I saw no books used in the classes (except collections of exercises and tables of logarithms), nor did I hear any allu- sion to the text in the work of the hour or in the assignment of home-work. One very excellent professor informed me, upon inquiry, that “Mehler ” was indeed officially in use, but that it was sometimes not alluded to in the class- work for months at a time. Whatever may be the extent of the actual use of the text-book, the German teachers are unanimously agreed on one point— purpose of the study of any particular topic in _‘ the text. the text-book must always follow the develop- ment of that subject in the class. The chief functions of the text-book are considered to be: first, to avoid the loss of time of instruction involved in the pupil’s copying the teacher’s explanations into a book as they are given. Second, to give those pupils who have not thoroughly understood the class presentation (and whose notes would probably also be faulty), a faultless presentation to which they refer and review the class-work, and clear up any points which had remained obscure. The books in use are accordingly written in a more or less brief form, as a skeleton, rather than as a complete body of instruction. Those 78 Teaching of Mathematics in Prussia books which do give detailed treatment usu- ally profess to do so out of consideration of the needs of such as may wish to use the book for private study, unaided by a teacher. Of course, opinions vary widely among the teachers as to the degree of detail with which a book intended for class use should be written. Those actually in use differ much in this respect, and many teachers publish, for the use of their own institutions primarily, supplementary and full- er treatments of particular chapters, topics, or class allotments than the concise text gives. IX The Examinations One of the results of the Conference of 1890, as embodied in the new curricula of 1892, is an attempt to diminish somewhat the The exam- tasks of the pupils. Among other __ mations. ‘things, the examinations have been made lighter. Annual oral examinations for promotion are held at the discretion of the Director and in his presence. So far as I could learn, these exami- nations are held on one or two subjects only in each class, and class-work alone determines the verdict in other subjects. The first formal examination comes at the end of Untersecunda and is known as the Adschluss- prifung. It covers the work of Untersecunda, and is conducted by a commission consisting of a Royal Commissioner, the Director, and the teachers giving instruction in Untersecunda. The questions are prepared by the teachers of the respective subjects and approved by the Director. The written examination in mathe- matics occupies four hours. In other respects, this examination is conducted like the final ex- 79 80 Teaching of Matbematics in Prussia amination, which will be described next. The certificate that the “ Adbschlussprifung” has been passed entitles the holder to a diminution of one year in the length of time of his mili- tary service (z.¢., he serves one year instead of two). The final examination, known as “ Rezfeprii- fung,’' or also as “ Abiturientenexamen,” is con- Thefinat | ducted by a Royal Commission, con- examination. sisting of a Royal Commissioner (who is appointed by the Provincial School-board, and is usually that one of its members who has the special supervision of the school in question), the Director, and the teachers giving instruction in Oberprima. Three months in advance the pupils give written notice to the Director of their desire to take the examination. The list of those applying is discussed in a conference be- tween the Director and the teachers in Oder- prima. The predicate to be assigned to each for his class-work is fixed (those in use are: Very Good; Good; Satisfactory; Unsatisfactory). By unanimous vote, those whose class-work has been unsatisfactory in all subjects may be excluded from the examination. The examination is both written and oral. 1In describing this examination, we give simply a sketch of its characteristic features, without mentioning the exceptions and alternatives which may arise and which are duly provided for in the regulations. The Eraminations Sr The subjects of the written examination and the time allowed in addition to the qhe written time mecaed to dictate the papers set® exemination. are as follows: German Theme, five hours; Mathematics, five hours; Translation from German into Latin, two hours; Translation from Greek and French into German, each three hours. Lexicons may be used, and in mathematics a table of logarithms. The mathematical paper contains four exer- cises, one each from Plane Geometry, Solid Geometry, Algebra, and Trigonometry. The papers set are afterward published in the Pro- gramm, and specimens will be given when the latter is described. In each of the subjects for examination— Latin, Greek, French, German, and Mathemat- ics—three examination papers are prepared by the teacher of that subject in Oderprima, and these papers, if approved by the Director, are sent to the Schulkollegium along with a list of the candidates and a sketch of their school- course. From each set of three the commis- sioner selects one to be used for the examina- tion and returns it in a sealed envelope to be opened at the time of the examination. He may reject all and prepare the paper to be used himself, if he sees fit. The papers are to make no requirements 82 Teaching of Mathematics tn Prussia whatever which exceed in kind or difficulty Independent the work of Prima, yet they must work differ sufficiently from the class-work required. to demand independent thinking; in mathematics, in particular, the problems must all be so-called “ originals.” If the papers written by the pupils are such that the Royal Commissioner doubts whether the pupils have really done independent work in the examination, he has the power to require a new examination, the paper being set by him- self. The writing of the papers is supervised by the teachers of the class by turns. The papers written by the pupils are first criticised by the teacher of the subject in ques- tion; all errors, even though very slight, are carefully corrected in red ink, and at the close the paper as a whole is characterized in a few words, and a predicate assigned by the teacher; the predicate for the class-work is also men. tioned. A few specimens will illustrate the character- izations of the mathematical papers; these gen- eral remarks are in addition to a specific predi- cate for each problem: “The paper of A. is correct throughout and skilfully written, and also quite extensive. Very good. His class-work in papers of similar kind was good. “B. has indeed made two gross errors, still The Eraminations 83 his paper may be called satisfactory in view of the ideas developed in it. His class-work in work of the same sort was satzsfactory. “C.’s paper is not free from obscurities and contains some mistakes, it is true, but it can still unhesitatingly be called satisfactory. His class- work has hitherto zot been satisfactory. “D.’s paper contains an oversight in the first problem, otherwise it is highly to be praised. Good. Class-work, good.” Next the papers are circulated for inspection among all the teachers of the Commission, who thereafter hold a conference with the Director and decide which pupils are to be recom- mended for exemption from oral examination. The corrected papers are then sent to the Royal Commissioner, who may alter the predi- cates. The oral examination must be held within the last six weeks of the school year at a date fixed by the Royal Commissioner who pre- The oral ex pigesseuctrcin. Iie determines the se- © s#aation. quence of subjects for this examination and the amount of time to be given toeach. All the teachers of the institution are required to at- tend the examination, from which are debarred those candidates whose written papers have all been “unsatisfactory,” and excused those whose written papers and class-work in all the required subjects have all been at least “ satis- 84 Teaching of Mathematics in Prussta factory.” Partial exemption from the oral ex- amination may also be granted, and according- ly only those pupils appear in the examination whose work has been unsatisfactory in part, but not entirely so. At most ten may be examined in one day. The questions in each subject are asked by the teacher of that subject in Oberprima and by the Royal Commissioner. In the course of the examination, the predicates to be given to each pupil in each subject are fixed by the Commission, upon motion of the teacher of the subject in question, and, in conclusion, the Com- mission holds a consultation under the presi- dency of the Royal Commissioner in which the final outcome of the whole examination is deter- mined. The pupil passes normally if none of his predicates in the various subjects (formed by combining the predicates for the class-work and the examination) are “unsatisfactory.” With certain restrictions, the predicate “un- satisfactory ” in one subject may be counter- balanced by the predicate at least “good” in another. This consultation is not a mere me- chanical collation of predicates, but when neces- sary the merits of individual cases are delib- erated upon as such. All the proceedings in connection with the examinations are fully re- corded. The pupils who have passed receive a certificate to that effect, containing all the Ube Lraminations 85 predicates and signed by each member of the Commission. The diploma of a gymnasium confers vari- ous privileges upon its holder, such as admis- sion to University study, to techni- priviteges of cal schools, to various examinations the graduate. for admission to certain military schools and the higher branches of the Government’s civil service, and is also accepted in lieu of some of the earlier examinations for officers in the army or the navy. Promotion to each of the four up- per classes opens to the pupil the door into some occupations which had been closed to him be- fore. Many of the privileges attained in the gymnasium may also be attained in the Real- gymnasium and in the Oberrealschule, though some of the most important may be attained in the gymnasium only. Xx The Programm The Director of each institution publishes annually. what is called the Programm, some- The scientific What analogous to our annual cata- papes logue. Itis usually in quarto form, substantially, plainly, and inexpensively gotten up. The contents fall into two main parts: A scientific paper (Wissenschaftliche Abhandlung) and the school-report. The scientific paper may treat an advanced topic, beyond the cur- riculum of the school, topics from the school- curriculum, or questions of pedagogic method. ~ It is not obligatory upon the incomplete insti- tutions to publish this paper. In 1896 there appeared 670 such papers in all Germany, the number of institutions being 993. The second part usually contains: I. The curriculum, in a table like that which we have used above. The schodl: IJ. The allotment of the hours of report. instruction to the various teachers, in a table which shows very clearly and readily the work of each teacher, and of each class. 86 The Programm 87 III. The Class-work.—Abstracts of the work of each class in each subject, with specification of number of hours per week and name of teacher. The abstracts in other subjects are given with about the same degree of detail as those in mathematics, of which we give the fol- lowing specimens for the four upper classes of two institutions taken at random. Humboldt Gymnasium, Berlin, 1896. IA. Mathematics, four hours. Voss. The no- tion of co-ordinates; something about conic sec- tions; maxima and minima; series; computa- tion of the known functions. A paper to be prepared at home each month. IB. Mathematics, four hours. Voss. Com- pound interest; extension of trigonometry ; imaginaries ; solid geometry ; mathematical geography. Each month a home paper. IJA. Four hours. Schnodt. Trigonometry; completion of the theory of similarity; con- struction of algebraic expressions; detailed review of powers, roots, and logarithms; equations (several unknowns) which can be re- duced to quadratics ; arithmetical and geomet- ric series; reviews in trigonometry. Home paper every four weeks. IIB. Four hours. Schnodt. Extension of the theorems on similarity; trigonometric 88 Teaching of {Mathematics tn Prussia functions in the right triangle; fractional and negative powers; logarithms; computation of the circumference and area of the circle; for- mule for the surface and the volume of the simplest bodies; equations of the second de- gree. Written papers every two weeks. Friedrichs Gymnasium, Berlin, 1896. IA. Four hours. Fischer, I. The number e- and the circle of numbers connected with it; supplement to solid geometry in connection with spherical trigonometry ; the notion of co- ordinates and the fundamental properties of the conic sections. IB. Four hours. Fischer, I. Imaginaries; combinations in application to arithmetical series of higher orders and to the binomial theorem; completion of trigonometry; solid geometry. IIA. Four hours. Fischer, I. Powers, roots, logarithms; arithmetical and geometric series of the first order with compound interest and annuities; conclusion of plane geometry; com. putation of the circle; plane trigonometry. IIB. Four hours. Schulze. Equations, in- cluding simple quadratics with one unknown; definition of powers with negative and frac- tional exponents; notion of logarithms; prac- tice in the use of logarithms; computation of Tbe Programm 89 the circumference and the area of the circle; the trigonometric functions; computation of right and oblique triangles; the simple bodies, with the computation of their edges, surfaces, and volumes. Besides these are given the subjects assigned for themes in the various classes and the ques- tions set in the written examination for gradua- tion. These questions are scrutinized by the teachers in other institutions, and in mathemat- ics at least form a fertile source of fresh ma- terial in the line of exercises for class use. We give now a few specimens of the final examination-papers in mathematics in institu- tions of the various types. Friedrichs Gymnasium, Berlin, Michaelmas, 1895. weet. Lhe fiye roots’ are to: be deter: mined both algebraically and trigonometrically. 2. A loan of 500,000 marks is to be ya sation- repaid by annual payments of 70,000 _ papers in marks. How many years will be re- "thematic: quired and how much will be the last payment? (Rate of interest not specified.) 3. In a triangle, there are given one side, c = 748m., the difference of the adjacent angles, d = 11° 29.4’, and the ratio of their sines as 4/3. The triangle is to be solved. 4. A segment of a material hollow sphere of 9o Teaching of Matbematics in Prussia radius r = 533cm. has the weight 1.5 grm. per qcem. of its surface. If now this~basin will barely swim in water (sinking to the rim), what is the depth of the basin and how large is the radius of the circular rim ? Humboldt Gymnasium, Berlin, Michaelmas, 1895. 1. Given a circle and an ellipse of equal area with major axis a. A rectangle is to be in- scribed in the ellipse equal in area to the square inscribed in the circle. 2. Given a circle and two tangents to it. They intercept upon a third tangent the length a. To find the angle between the third tangent and one of the others. To be worked first in theory and then computed for the following numerical values: r= 6.082, a = 35.025, and the angle between the two given tangents Oa oh 3. A body of weight q falling from height h with initial velocity c penetrates s meters into the ground. How large is the resistance of the ground? To becomputed also numerically for h = 347m., C= 7m., S = 0,65m., and =e 4. About a given spherical segment a cone of minimum volume is to be described. Tbe Programm Ol Luisenstadtisches Realgymnasium, Berlin, Easter, 1895. 1. What conic section is represented by the equation (rectangular co-ordinates), 4x? — 4ryt+y? — 64V 4 — 87/442 Oe 2. Solve the system KU = xKE=10; e+y—uU—2=4; e409 +y2 +2? = 130. 3. In the geographical latitude of Berlin, what time lapses on the longest day until the middle point of the sun is 6° below the horizon? 4. Given the radius r and the altitude h of a right cone, to determine the cylinder of maxi- mum volume which can be inscribed in it. Friederich Werder’ sche Oberrealschule, Berlin, Easter, 1895. I, From a triangle with sides a, b, c, a rec- tangle is to be cut such that the cylinder formed by bringing a pair of opposite edges together by bending a rectangle shall be as large as possible. 2. The sinking sun shines into aroom through a circular opening in the front wall,and throws an elliptic spot of light on the side wall. From what point in the rear wall does this patch of 92 Teaching of Mathematics in Prussia light appear to be circular? Construct this point, if the floor-plan of the room, the position of the circular opening, and the direction of the sun’s rays are given. 3. By what equation is the radius of a sphere determined on which a triangle formed by three arcs of great circles has sides a, 2a, 2a, and the angle A between the equal sides? 4. Investigate the curve 4x? — Oxy +4lye+ 360% — 52v+10I =0, and compute the area enclosed by it. Eighth Realschule, Berlin, Easter, 1896 (Aé- schlussprifung—te., at end of Untersecunda). 1. A wooden sphere sinks in water to one- half its height and in alcohol to 7/12 its height. What is the specific gravity of alcohol? 2. On the side AB of a given quadrilateral ABCD arectangle is to be constructed equal to the sum of the three squares on the other three sides. 3. In an arithmetical series, the sum of the second and the eighth term is 22, and the prod- uct of the third and the sixth term is 91. What are the first term and the difference of the series ? IV. List of Text-books Used.—Sometimes these are given in connection with the class-work The Programm 93 (Pensen); sometimes no information on this point is given. The list is often in the form of a table making clear to the eye at once what books are used in each subject, in each class, in what classes each book is used, and the price of the book. The use of text-books in mathematics has already been discussed. V. Orders of the superior boards, so far as they are of general interest. As specimens we give the following, taken from the Programm of the Luisen Gymnasium, Berlin, 1896. When not otherwise specified, the orders are from the Royal Provincial Schulkollegium. 1895. March 18. Physician’s certificate required with appli- cation for excuse from physical train- ing. neat. The 80th birthday of Prince Bismarck is to be a holiday. April 3. Arrangement of vacation courses in natural sciences and archeology. Dr. Fritz Bosch assigned to the institution as cand. prob. May 2. Days on which the institution has to flag. araneO. Approval of the rules of the boating di- vision, June Io. Tickets are received for five pupils and one teacher for the Luther Celebration in the New Market. ea To The work of Lindner on the war of 1870-71 is recommended, 94 August 2. ak ES « A ae et uirert & See Lae October 17. November 1. ce 2. é¢ 8, “6 19. “6 26. December 10. 6e¢ ve «¢ 19. Teaching of Mathematics in Prussia Danger of firearms in the hands of pupils. Approval of the leave of absence of Professor Dr. Weber. The institution is to participate in the Sedan celebration in the Lustgarten. Freiherr v. Mirbach sends from the Civil Cabinet of Her Majesty the Empress an invitation to the consecration of the Em- peror William Memorial Church. To illuminate on September 2d (Sedan day). Permission to the institution to participate in the consecration ceremonies (October 21st) of the Emperor Frederick Memorial Church. The vacations for 1896 are fixed. The introduction of the new French school- book by Ploetz Cares is allowed. Permission to add a seventh hour of Latin in Prima and in lieu thereof change one hour of physical culture into open-air games. Notification that Professor Ewald, Director of the Art School, will inspect the instruc- tion in drawing. The Stenographic Association is allowed the use of a room. Programme for the celebration of the 25th anniversary of the German Empire, on January 18, 1896. Addresses, songs, dec- lamations. Professor Dr. Gemss and Professor Dr. Weber receive the rank of Councillors of the Fourth Class. The Minister presents the work of Breysig: Brandenburg sche Finanzen. The Programm 95 1896. January 17. From the administration of the privy purse (Schatullenverwaltung) of His Majesty the Emperor the institution receives as present the picture “/Vations of Europe,” by His Majesty the Emperor, with auto- graphic signature. af $e Three copies of Lindner’s book, the war of 1870-71, are sent for presentation to good pupils; also three copies of the address of General von Mischke at the unveiling of the Emperor Frederick monument near Worth. fy Leave of absence granted to Professor Dr. Weber. ae a To illuminate on January 18th. ss 28. The workof Réchling and Knotel, Der alte Fritz, is commended. February 6. From Easter on, the Jewish religious instruc- tion is to be combined with that of another institution. 27. Wall-charts recommended. The above are about half of those published in this Programm as of general interest, and il- lustrate well how thorough and detailed is the supervision by the Schulkollegium. VI. Chronicles of the Institution.—This is a concise sketch of the life of the institution. Festal days celebrated are briefly described; occasionally an address by the Director is pub- lished in full; excursions made are enumerated; omissions of instruction for any reason specified in detail; changes in the Faculty are recorded ; 96 Teaching of Mathematics in Prussia if new members are appointed a full account of their previous career is given; death or serious cases of illness among the Faculty or pupils, as well as any distinctions or honors that may have come to any of the Faculty, are suitably noticed; visits of inspection by the Provincial School Councillors (members of the Schudlkolle- gium) are described in detail, with words of thanks for the advice and encouragement re- ceived; in short, any deviation from the ordi- nary routine of the school is here made the subject of concise record, so that on one or two pages a clear picture of the school-life is painted. VII. Statistics—1. Summary of the attend- ance, by religion and residence. 2. Table of the attendance in the various classes, gain or loss during the year, and the average ages. 3. List of the graduates, giving for each, name, place, and date of birth, religion, years in the institution, years in Prima, occupation of father, and prospective occupation of the graduate. VIII. Additions to libraries, laboratories, and museums during the year. These are enumer- ated in detail, and all gifts, however slight, are mentioned in connection with the name of the giver and words of thanks. IX. Stipends and funds for the support of pu- pils. A report of receipts and expenditures, The Programm 97 X. Notices to Pupils and Parents.—Information regarding the rules and administration of the school. Under this head almost all of the Programms in Prussia in 1896 brought in full a long, cautionary letter from the Minister of Education concerning the danger to pupils in handling firearms which was called forth by the accidental shooting of two pupils in this manner somewhere in Prussia. The information contained in the Programm is almost invariably grouped under the above heads, but their order is sometimes varied. XI The iReformschule In what has preceded, the work of the Prus- sian higher-school system has been described in so far as is necessary in order to understand the character of the work in mathematics done in these schools, but before closing this descrip- tion a few words may be added concerning two topics of some interest in this connection, the Reformschule and the higher education of women. In several institutions known as Reform. schulen the experiment has recently been inau- Character gurated of building curricula, in- and purpose. tended to be equivalent to the three now current, upon a common substructure up to the close of Untertertza, and of not separating the gymnasial and the realgymnasial course before Untersecunda. These institutions aim primarily to defer the decision between Greek or no Greek, Latin or no Latin, to a later period than the present curricula permit. As matters stand at present, the parents of the boy must decide which of the three classes of institutions 98 The Reformscbule 99 he is to enter when he is only nine years of age, and when once the boy has made any progress in the work of any one class of institution he can transfer to another class only with consid- erable disadvantage. In the Reformschule, the choice between Latin and no Latin need not be made until the boy is at least twelve years old, and the choice between Greek or no Greek is then deferred two years further. To accom- plish this and still give an equivalent amount of Latin and of Greek it is necessary to give eight hours a week each to Latin and Greek during the last four years of the gymnasial course, and ten hours a week to Latin during the two years preceding these. The total amount of time given to Latin and Greek under this plan is not so great as that given under the present curricula, but it is believed by the advocates of the plan that the work done under the two plans will be equivalent, because the pupil should be able to accomplish more per hour in the latter part of his course than in the earlier years. To counterbalance the Latin and Greek thrown forward, French is thrown back into the earlier years, and more hours are allotted to it than under the present plans. In how far these and other expectations of the friends of the new movement will be realized remains to be seen. The ministry gives these institutions free scope for the trial 100. = Weacbing of Mathematics in Prussia of the experiment, and the outcome is awaited with much interest. The first institution to begin the experiment was at Frankfurt a/M., where the work was begun in 1892; since then one institution each in Hanover, Breslau, and Berlin have in turn taken up the plan. As these institutions had previously been working on the usual curricula, the transformation of the institution into a Re- formschule-must be made gradually with the progress of the classes, those which had begun work under the standard curriculum having to be carried through according to the same. The process of transformation consequently re- quires nine years for its completion, and the first Reformschule, that at Frankfurt a/M., will graduate its first class as a Reformschule in 1901, and the other institutions several years later still, As no institution is as yet working under the new curriculum solely, it is too early to speak definitely of the outcome, and we close with the detailed curriculum of the Lezbnig Reformschule in Hanover, and a comparative summary table of the curricula at Hanover, Frankfurt, and the current plans of 1892. The Reformscbule IOI ,CURRICULUM OF THE REFORMSCHULE AT HANOVER. Middle Substructure. era eure WE Vi IV. TTB TLEAS SSeS (Ce ey | Chrrstian Religion... 5... ...,.5-5 3 2 2 2 2 ERMIR ES as » History and Geography..... 28 24 26 29 27 Mathematics 5. 05. C72 saan ce 38 37 34 43 42 Natural History............ 10 10 8 10 10 POP UCE Pes cas a eae Ee ay 8 8 Io 12 9 Chemistry and Mineralogy. . as 3 be 6 6 AV Siting «iss ces egieoraceeeee 6 6 4 6 6 Drawing 5 nee fees sae tre 8 8 8 16 16 MOA Joie Paces en deere 264 257 252 266 262 259 XIT The higher Loducation of Women But little has as yet been done toward pro- viding for women similar educational facilities to those for men which we have just No gymnasia been describing. Nofull-fledged gym- fer women. nasia for women exist, and the proposal to found one in Breslau was vetoed by the Minis- ter during my stayin Berlin. In the latter city an arrangement has, however, been effected _whereby the “Courses of the Gym- What nasium’” (Gymnastalkurse) may be is done. taken by women under the instruction of vari- ous strong professors from some of the Berlin gymnasia. The courses are given in the afternoon when the instructors are free from their regular du- ties, and opportunity is given to those women who have completed courses equivalent to the curriculum of a Gymnasium to pass the regular final examination before some one of the Royal Commissions conducting the examinations of the Berlin institutions. The women who pass 103. 104 Ucaching of Mathematics in Prussia the examination successfully are admitted to University lectures under certain conditions. This work for women is under the direction of Miss Helene Lange, whose strong plea for the privilege of higher education for women is known to American readers in a translation published under the title “ Higher Education of Women in Europe.”! Through the courtesy of Miss Lange I was permitted to attend several classes in mathe- The quality Matics,and found the young women ofthe work. doing excellent work. As might be expected, they were distinctly more mature than the boys doing the same grade of work, and were decidedly more earnest and serious in their work; the social conditions in Germany are still such that a woman who seeks a higher education is regarded as a pronounced “ blue- stocking,” and this state of affairs deters all but women of fixed purpose and strong character from taking up these courses. With all the social traditions and prejudices discountenanc- ing this form of education for women and just as strongly urging it upon men, and even mak- ing it absolutely prerequisite to nearly all of the most desired careers, it is not surprising that, while boys of all degrees of talent flock to the gymnasia, only women of marked strength 1Lange, Higher Education of Women in Europe, D. Appleton & Co., 1890. The thigber Education of Women 105 are brave enough to carry through the corre- sponding work. Miss Lange has pointed out in the book men- tioned above that Germany is behind the other European nations (not to mention progress America) in the paucity of the facil- being made. ities for the acquisition of a higher education which it offers to women, but progress is being made, and year by year the privileges of women in this regard are being increased steadily, even though slowly. XIIT Comparison between German and American Work We are now ina position to consider in detail the facts upon which was based the assertion made at the beginning of this report—vzz., that in the corresponding nine years we Americans accomplish no more in mathematics than do the Prussians, and that we use up seven-fourths as large a fraction of the time of instruction in doing tt as do the Prussians. As the American basis of comparison the Chicago schools (Grades, High-Schools, and Basis of the University of Chicago) were comparison taken, in the belief that they are on the whole as typical of the better class of Ameri- can schools as any concerning which facts could readily be obtained, and that their work is at least up to the average of that done in insti- tutions of similar character throughout the country. The nine years in the Chicago system which correspond to the nine years of the gymnasial course are readily seen. 106 German and American Work 107 At the age of nine the boy enters the gymna- sium, and here at theageofnine he g, nos enters the fourth grade and has had American something of fractions, decimals, and Sere denominate numbers in addition to what the German boy has had. He continues arithmetic during the next five years to the close of the eighth grade, but a part of the time allotted to mathematics in the latter grade is devoted to an introduction to algebra. In the first three years of the High-School, algebra through quadratics, plane and solid © geometry are taken up; during the fourth year of the High-School no time is given to mathe- matics, and consequently the first year in college constitutes the ninth year of mathemat- ical work; in this year plane trigonometry and college algebra are given twelve weeks each. The nine years of American school-work which are thus brought into comparison with the nine years of the German gymnasium are the five years next preceding the High- School, the first three years of the High-School, and the Freshman year in college. In the last year the American has one year’s advantage over the German as to age and mental develop- ment, but he has also the disadvantage of tak- ing up his work in mathematics after having suspended it for a year. 108 Teaching of Mathematics in Prussia If the German curriculum as given above be compared with the ground usually covered Workdone in the nine corresponding years in compared. America, it will appear that the topics taken up are in substantial agreement, with the exception of the conic sections, something of which is contained in the German curriculum but not in the work of the American years in question. The text-books used and the exercises in the classroom both indicate that the Ger- man treatment of the topics is, on the whole, at least equivalent in scope and thoroughness to that in America. The test which the reader can most readily apply in this respect is to scrutinize the examination-papers which the German boys pass at the close of their work. The specimens given above may be regarded as sufficiently characteristic. Of course the German pupils have been gre- pared to pass such papers, even though the questions are all supposed to call for inde- pendent thinking at the time of the exami- nation, but perhaps the chief object of final examinations of this character is to test and exhibit the extent and thoroughness of the preparation of the pupils. There is little ques- tion that the German pupils would be able to pass the corresponding examinations given here. German and American Work 109 The following table taken in connection with the preceding description of the Prussian gym- nasium, will substantiate the assertion Time ratios which was made at the outset. It compared. should be noted first that the comparison is made between fractions of the total time of in- struction and not between number of hours of instruction, for if four hours’ instruction out of eighteen weekly be given to mathematics, more work should be accomplished per week than if four hours out of thirty are given. The smaller number of hours per week instruction implies more time for private or home-work, and math- ematics is sure to secure its full share of this time. In fact, it is very possible that the aver- age American pupil gives to mathematics more than its proportionate fraction of his private work. In Germany, the allotment of timeto the various subjects, the record of assignments for home-work, and the methods and traditions of instruction all combine to make it improbable that more than the allotted portion of home- work accrues to mathematics. The table will be understood without further explanation. Inthe statement of the fraction of the total time of instruction given to mathe- matics the denominator is the total amount of instruction per week, and the numerator is the amount given to mathematics. 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