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ested in the thought development of today can hardly help asking about the fourth dimension when some bright boy will ask for a figure to represent it.
Then comes the statement, mysterious to the pupils, that we cannot see the geometrical concepts. By telling them that this is not new to them, that they have always worked with invisible concepts, the class will be turned into an interesting guessing party. All sorts of answers will be given until some one will think of figures as representing numbers—then words repre senting thought-letters-sound-, and so they can be led on until for the first time they discover that even the body is but a symbol of the real self. Right here the teacher has a wonderful opportunity for a life lesson that the pupil will remember long after he has forgotten Geometry.
The second day we discuss the angle, which is as difficult for the pupil to understand as it is for the Mathematician to define. This definition may be illustrated by having students walk from a corner of the room making different angles with the side of the room, or by the revolution of a pencil from its first position about one extremity, or by using the points of compass, from which the pupil will get the number of degrees in a right angle and also in its parts.
The definition of a theorem is quickly understood by calling it a conditional sentence. The pupil easily makes application of this selection the Hypothesis, and the Conclusion in several theorems.
We find the self-evidence of the Axioms depends upon the self, and we must patiently work even with these until the pupil understands how to apply even one so simple as, “Things that are equal to the same thing” etc. He does not understand why the same thing is not included in the conclusion. Often the pupil will quote this axiom without knowing what the same thing is, or what the two things are. In the Axioms, “The whole greater than any of its parts,” and “The whole is equal to the sum of its parts” can be given an appeal for best scholarship and best citizenship which is always needed and appreciated.
The first theorem we have is: Two triangles having two sides and the included angle of one equal respectively to two sides and the included angle of the other are equal in all respects. The day it is assigned as the lesson I repeat it once and ask the class to repeat it in concert—then ask for the separation of the Theorem into the Hypothesis and the Conclusion. Then I draw the figures, carefully constructing according to the hypothesis-telling the pupil to observe closely so he can do it on the following day. The pupils enjoy the accurate drawing and all that is necessary in teaching the pupil to make angles equal, is to tell him to make the lines the same distance apart at the same distance from the vertex. I then give the demonstration, after which I call upon some member of the class to give it. There will be several in the class who can do this, while from lack of imagination others find superposition very difficult; for these we draw two polygons and then apply one to the other. They will learn before completing this that the direction of a line from a given line depends upon the size of the angle made with the given line—and that the location of the next vertex depends upon the length of the side.
Theorems one and two are learned by oral lessons—the pupil not reading the demonstration in the book. They are warned from the first to learn the theorems exactly as given but to avoid memorizing demonstrations. From the very first the pupil takes his place at the board standing where the class can see the figure and learning to use a pointer by pointing not at but to a letter. This accurate pointing helps the one reciting and it is a great aid in holding the attention of the class and, also, in making it easier to follow the demonstrator.
The application of the hypothesis, the application of the conclusion, the demonstration proper, the student will gradually learn to give in the right order, but, during the first two or three weeks, the teacher will have to be patient and constant in giving suggestions.
No mistake is too small to be corrected as it is best to insist upon accuracy from the beginning even in naming lines in the right direction.
We all recognize the difficulty in preventing memory work as the pipil can make a better appearance before the class by reciting something he has memorized rather than by thinking his way through the demonstration. The pupil is given plenty of time at the board to think.
The teacher of Geometry must constantly keep in mind that education is not learning but is exercising the power of the mind. The pupil should learn this early and more praise should be given for the simplest original demonstration than for the most complicated one which is given to him. Early, also, should he learn the use of the syllogism and so hunt for the two statements which put together will give him a conclusion leading in the direction of the result, then hunt for another statement to put with this conclusion until he can make another conclusion and so on until the theorem is proved. The pupil must see that a demonstration is like a tree in which leaves, twigs, and barnches grow out of what precedes.
Direction of thought, choice of truths leading in that direction, and combining them to the right end, these should be brought early to the attention of pupils and put in practice. This I try to do by telling them to consider, every time they prove a point, how the statement is to be used. Why is it true? Why did I prove it? Where do I use it? A large number of simple treorems on proving triangles equal is of great value to the pupil.
Often a pupil understands a principle but fails in applying it to the geometrical concept. For instance, in the demonstration of "At any point in a straight line only one perpendicular can be drawn to the line which shall lie in the same plane with the line," two angles are proved unequal by comparing them with angles that have been proved equal. This principle we illustrate the day the theorem is assigned by having four boys A, B, C, and D, stand before the class, B. and C. being of the same height, A. taller than B. and D. shorter than C. The class will easily conclude that A and D are unequal in height. Tell the pupils to look for this principle in the lesson for the following day. All will be interested and there will be few failures. This third theorem introduces the pupil to the indirect proof. He will understand the simple statement if this is true that is also true, that is not true therefore this is not true. If the pupil early gets the correct idea of the indirect proof it will be of great asistance to him especially if the author uses many of them as does Sanders in six of the first eighteen theorems. The pupil should be able to give a description of this proof as suppose the conclusion false and something else true, continue the argument until a statement is reached which contradicts a previously proved principle, a definition, an axiom or the hypothesis in the theorem, then it must follow that the first supposition is false and therefore the theorem is true. I insist upon the statement which contradicts.
Our theorem four is based upon the equality of triangles. Let the pupil understand that triangles are always proved equal for a purpose—either to prove two angles equal or two lines equal. I give them the word homologous early, to avoid confusion,
If at times there is a lapse of memory in regard to some mathematical principle or operation that it seems to us that pupils could not help knowing let us not find fault with him but stop right there and help him to recall that with which he was once familar. If we are going to assist the pupil, we must do it at the place where he needs assistance. Ideas take possession of us according as we dwell upon them and use them. First I try to be sure that the pupil has a clear conception, then hope that eternal repetition may fix the principle in his mind.
The student must understand that a demonstration is like a rope ferry with one end of the rope attached to the hypothesis, the other end to the conclusion and he must work his way from the first to the last, that is he must pull thru the demonstration. Of course from the first good English must be insisted upon.
Often a pupil makes a correct statement without understanding it. This is for the watchful teacher to discover and correct by questioning In such a case we often hear from the pupil “I knew it but you confused me by your questions."
By asking the pupils to study, a part of the recitation hour, we will see that they are saying the words over and over-that is-studying mechanically. This, while of value in some subjects, is fatal in Geometry.
The boy or girl is often deceived in his own work and is unconsciously memorizing. During the first two weeks the studying except the memorizing of theorems should be done in the class room. There is a place for memorizing in Geometry and in that place it must be thoroughly done. I insist that as the pupil recites a theorem he must have in mind the figure which he is describing. He must always visualize the figure or he can not do the work-my practice is often to ask the class to hold in mind the figure while different ones give in order the principles used in the demonstration. This requires concentration. Schulte says, “The teacher must guide the pupil into the slow, judicial method of study. This excessive use of memorizing and the neglect of the cultivation of the reasoning power are possibly the worst effects of the spectacular idea upon which our schools are largely built."
The slow, thoughtful recitation is to be encouraged. Never hasten the pupil who is thinking. In Geometry the data are few and definite; the reasoning simple and accurate; the results certain. There is no chance for personal opinion. The pupil learns to know when his work is right. Clearness and exactness are necessary in life. These, Geometry should aid the student in gaining. The pupil soon learns that the subject is not informational but that power is the end in view. We are all familiar with the bright happy face which shows that the pupil has discovered that he has within himself the power to take the data and work to the conclusion. Many difficulties there are of which I have not spoken but let us teachers not be too anxious, knowing that when our duty is conscientiously done, we have nought to do with results, for we are only the instruments thru which the Universal works.
PROFESSOR WALTER FRENCH, MICHIGAN AGRICULTURAL COLEGE
The subject of Botany may be considered as the elementary science of our secondary courses. It is at least the basic science for the subject of agriculture. Its place in the course of study may be debated because it is probable that pupils in the 9th grade have not sufficient maturity to derive the largest possible benefit from the subject, but because of its basic character it will in my judgment be necessary to introduce the subject at that point in the crriculum. In the preparation of this paper I have received valuable suggestions from Professor Bessey of M. A. C., and Professor Hummel of California.
I do not think it necessary to make any very sharp distinction between Botany and Agricultural Botany, for the whole subject, as treated in secondary schools, should be considered a unit, and the agricultural phases presented and discussed as they appear in the progress of the work.
The character of the instruction in Botany in the secondary schools should vary somewhat according to the location and environment of the school and the student. If a school is located in a part of the state where horticulture is the dominant industry, then the student of Botany in that place should give particular attention to the subject of plant propagation as related to the plants cultivated in that community.
In the suggested course in agriculture as presented for the schools of Michigan, we specify Botany and Agricultural Botany, but we do not mean by this necessarily that the first half of the year shall be given to pure Botany and the second half to agricultural application. We believe that because of the basic character of the subject it should be given an entire year, and the instructor should recognize the environmentta conditions, as above stated, and he should also adjust the work according to seasonal conditions and vegetation which may be studied. For instance, in the fall when plants are fruiting, the student should be given opportunity for study of the text and practical observation of fruits; while in the spring of the year the subject of plant propagation, seed testing, etc., should be given attention. If the botanical laboratory is properly stocked with preserved samples, the seasonal conditions will not affect the subject so materially.
The first work of the student will be devoted to the structure and function of plants, also a general sketch of the whole vegetable kingdom, so that the student shall secure a vision of what Botany means. This work will be followed by learning how to identify plants and how to use a botanical manual by actual observation and experiment work in the laboratory. During the fall and spring months it will be profitable for instructor and student to have laboratory exercises out of doors in connection with plant life in its actual conditions, as to habitat, manner of growth, manner of reproduction, and practical use.
It is my opinion that the first few weeks of the freshman year should be used by the instructor in science in giving to the student a hasty glance of the general fields of science. This can be done by means of lectures and simple illustrations and experiments touching upon the different phases of science. This will arouse the interest of the student and show him the possibilities of future study. I believe this is a vital thing in connection with science in our secondary schools, and the instructor should arrange to give to the students a series of demonstrations touching upon fundamental terms and processes in Botany, Zoology, Physics and Chemistry. It is not necessary to go into detail as to what the instructor should do.
For the purpose of this discussion we shall consider that the regular instruction in Botany will be planned by the instructor to fit into the agricultural phases. Then I would make the following suggestions on the agricultural side: