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as some of ours. Of course, German spelling does not present the difficulties of the English. If one knows how to pronounce a word, he can usually spell it; so this saves a great deal of trouble to the teacher, and mistakes seldom occur. Spelling by sounds, or analyzing words, I have heard here very thoroughly taught. In drawing, also, the pupils are made to be very accurate. They draw with pencil and crayon, mostly with the former, but the style of the finished pictures is scarcely ever as bold and free as with the scholars of our best drawing teachers.
The class invariably rise when the professor or a stranger comes into the room, and remain standing till permission is given them to be seated. I do not see, otherwise, any great difference between them and American children in school manners. The classes are never so quiet as in our schools, and the noise and whispering do not seem to trouble the teacher very much.
There are in Berlin very many of what are called Kindergarten schools, to which the youngest children are sent before they are old enough to enter the larger schools. These are conducted after Froebel's plan, and only in these schools, so far, are his ideas adopted. But his intention was that the same principles which govern the management and teaching of these schools should be carried on still further into the higher ones. Pestalozzi is followed here, but one great idea of Froebel's was that the child must be taught himself to do with his hands what he sees; that he must first see and understand and afterwards do it. For their hours of recreation they have spades and rakes and hoes and wheelbarrows, and each a little garden plat, and they are taught to use all these things, to plant and care for flowers and vegetables, and to watch their growth and development. Perhaps it is hardly to be wondered at by those who know the opposition which manifested itself when it was proposed at Harvard to extend the course of study in the natural sciences, that there is considerable objection to these schools on the ground that they do not teach enough religion, and that their tendency is anti-Christian. For “religion "please read “ theology." Every school here must have a clergyman connected with it, and these clergymen are sometimes much shocked at finding that the little child of three years, whom he finds occupied in cutting a cross
from paper for a birth-day gift, is not able to tell him of what the cross is a symbol, and persists in declaring simply that it is for a book-mark and is intended “for mamma." This seems, to say the least, unreasonable, but in a state where religion must form a part of the regular study of every school, and where it stands on the same ground as arithmetic or geography, with the only difference that the hour devoted to religion is more tiresome than perhaps any other, it can hardly be wondered at. To an American it seems as if everything was fettered here, and as if the people must chafe continually like a high-spirited horse with a cruel curb continually irritating him. It seems as if they were allowed no liberty of thought, even on those subjects on which no one should dare to come between the soul and its Maker, and one understands readily how the girls who are thus trained from earliest childhood, never throw off the mental clothing which has been so closely fitted to them, while the boys, in withdrawing themselves eventually from parental restraint, shake off at the same time all restraint, and cast aside as fables and idle tales the doctrines which have been so tirelessly taught to them in their childhood. All these regulations are for the safety of the kingdom and the king; but one continually wonders—and especially now when we can almost hear the roar of the cannon which may throw all central Europe into a fearful struggle, and entirely change the condition of Germany, how long this will last, how long it will be before all the misery and suffering which lies concealed behind the splendor of royalty will come forth and claim its right to liberty and the pursuit of happiness.
I cannot close without speaking of a Geography on which Mr. Theodore S. Fay, for so many years our minister to Switzerland, has been engaged for some twenty years. Mr. Fay leaves Europe for America early in July to publish the book and to make an effort to introduce it into our American Schools, so that you will soon have an opportunity of judging as to its merits. I have seen the maps which are to accompany it, and which have been most carefully corrected and approved by the best geographical authorities here, and can most sincerely say that they are the most perfect and thorough I have ever seen. No labor or expense has been spared in their preparation and correction, and when they are pub
lished they will be as exactly correct, even to the smallest particulars, as the knowledge of man can make them. They contain many new features with regard to mathematical geography, as to length of days, change of seasons, &c., of which every teacher will be glad to avail himself; and the “voyage map," as it is called, on Mercator's projection, is destined to be a source of endless pleasure and profit to both teacher and pupil. The text which accompanies the maps is most carefully and thoroughly prepared. I am certain that the excellences of the book must commend it to every teacher. It will save an incalculable amount of labor and trouble, and fix all the details of geographical knowledge so firmly that the child will no longer need a map for reference because he has the knowledge in his mind. I shall impatiently wait for its publication, knowing, as none but those who have taught geography can know, how much time and patience it will save; and I am sure that all American teachers will give it a hearty welcome as a help in a direction where so much help is needed.
A. C. B.
ON TEACHING GEOMETRY.
My method may not be new to all, but it may suggest something new to a few. It has been wrought out during several years of teaching, and every part has been thoroughly tested. The results of the trial in my own classes, composed of girls sixteen years of age and upwards, are far superior to any I could obtain by the use of old methods. It is always hard to make new plans, - plans in which we have confidence. There must be time to try many methods, to digest the results of experience, and to rearrange material. And since I know how much more may be done provided one has a good plan in the beginning, I venture to speak of my method, for the sake of those whom I can help, though it may contain nothing new to many of my readers. The evidence that there are those who need help accumulates with every new class that comes to me for instruction. I find many who have studied the subject before, but in such a way that the memory alone has been exercised, and that to a merciless degree. A theorem is demonstrated as a list of
prepositions might be repeated. Certain words in a certain order, are learned mechanically. There is no understanding between the mind and the eye as to the figure upon the board, except so far as the letters, placed in exactly the same order as they were learned in the book, aid in recalling the words. Change the letters, turn the figure upside down, or ask for any particular step in the reasoning and the recitation is stopped. That Geometry has anything to do with reasoning, or is a discipline for any faculty except memory, has in many cases never occurred to the scholar. It is a hard thing to say, but its truth has been proved.
• With my first class I experimented, trying to ascertain what had been the exact method by which they had been taught. I changed the position of the figures, changed letters, asked questions which showed the class there were such things as relations between facts, that one thing followed from another, and I found that they had no conception of the nature of the subject. For that term, I allowed them to go on, demonstrating with the aid of the figures, and labored to teach them to reason. The disgust with which they soon regarded the old ways, and the pleasure which they felt when they could put two things together, and see the relation, and how a third grew out of a combination of those two, amply repaid me for the work. But the results in all cases were not quite satisfactory.
With the next class, I varied the method somewhat. After going over the first book for instance, using the figures, sometimes with letters, or the Arabic signs, and sometimes nothing but such words as were appropriate, the class reviewed, demonstrating every theorem without a figure. I required that it should be done so fully and carefully that a person listening should not feel the need of a figure. Take Prop. XIX., Book I, of Davies' Legendre:- If two straight lines meet a third line, making the sum of the interior angles on the same side of the line met, equal to two right angles, the two lines will be parallel.
Through the middle of the secant line draw a line, perpendicular to the lower given line, and prolong it till it meets the upper given line, thus forming two triangles. By Prop. I., the angles formed by the secant line and upper line are equal to two right angles. By hypothesis, the two interior angles on the same side of
the secant are equal to two right angles. By axiom 1, these two sets of angles are equal to each other. By axiom 3, taking away the common angle, the remaining angles are equal. These angles are corresponding angles of the triangles. The angles of the triangles, formed by the secant and the proof line are equal, because they are vertical. By construction, the sides which are parts of the secant are equal. By Prop. VI., the triangles are equal. By construction, the angle formed by the proof line and the lower line is a right angle. Its corresponding equal angle must then be a right angle. Therefore the proof line is perpendicular to the upper line, and conversely, the upper line is perpendicular to the proof line. By Prop. VIII., the two given lines are parallel.
When a scholar can give a demonstration like that, she knows it; she can commence in the middle as well as at the beginning; she can tell you the reason for every step; and with all this exercise in reasoning, she has the benefit of the thought necessary to put it all in concise, clear language. The lesson proves to be one in many things.
Accuracy in habits of thought is to be one of the results of the study of mathematics. The reasoning must be accurate, that the conclusion may be correct. Carrying out the principle, the language should be accurately used, and the order of the parts of a demonstration accurately observed. It has been my practice to give first the theorem in general terms, then the application to the figure in question (when using figures), making that application in the exact words of the theorem. I would banish the word "let" entirely. Thus, applying Theorem I. Book I., “ If the straight line A B meet the straight line C D, the sum of the two angles A B C and A B D will equal two right angles." The theorem is the general statement, the application the particular. Why should we say, “ Lct the straight line A B meet the straight line C D, &c.”? It seems a trifle, but as part of a system it becomes important. Next comes the preparation of the figure for proof, all proof lines dotted, and lastly, the proof itself.
The method of demonstration without figure develops the power to reason upon each separate theorem. The facts of Geometry are thus learned in detail. The next step is to group these facts