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for Geometry as compared with that shown for Arithmetic. At first it occurred to me, that this might be due to the fact that the study of Geometry was new; but as time went on and this interest remained undiminished, I could only ascribe it to the fact that the study was more adapted to the mental development of the boys.
But a second fact worthy of attention, may be recorded. This class of boys, who as a rule were poor in Arithmetic, and found great difficulty in comprehending arithmetical reasoning, seemed to thoroughly enjoy the study of Geometry. They followed the course of reasoning rationally and with understanding, they were able and eager to reproduce it, they would detect and correct flaws in the reasoning of others. These facts gain a greater significance from the following, that in no case were they allowed the use of textbooks. Nor was the advance in the class confined to those pupils who had the best record in Arithmetic. On the contrary, some of the boys who stood lowest in Arithmetic were the most apt in Geometry, and there was no really poor pupil in the class. Such a progress in the line of mathematical induction was not without its good results, for as the reasoning faculties were thus increased, there was a very noticeable improvement in the class in Arithmetic.
Another significant fact, upon which perhaps the greatest stress may be laid, was this: that the study of Geometry seemed to awaken within the pupils the power of applying material, already acquired, to a proper end, and of thinking independently. This should be the ultimate and highest aim of all educational efforts. One instance out of many, as an example, will show what I mean. The class had just proved the proposition, that the angles opposite the equal sides of an isosceles triangle are equal, whereupon I asked, if any could prove that all the angles of an equilateral triangle were equal. After some thought, several hands were raised, and I called upon one of the youngest, a boy of eleven years, to attempt the proof. He promptly responded, carried out the proof logically, applying correctly the former proposition, and without any prompting. Here we have, then, several practical results of importance, a class of young boys studying Geometry with great interest and understandingly; in consequence taking a greater interest and improving correspondingly in Arithmetic; showing a mental development and a growth of the reasoning faculties, which hardly would have been attained otherwise. I felt
confident before, I feel sure now, that good results may be reached in this way, that the hostile attitude of many toward Mathematics may be entirely changed, and that the mental development will be
As to the question of the proper time when to begin the study of Geometry, it may be considered open. Geometry can be introduced, as I have demonstrated, at about the eleventh or twelfth year, possibly sooner, and for a certainty the study of fractions should not be taken up, until some such training has preceded it.
I come now to the consideration of how the study of Geometry may be presented to the pupils at the early age mentioned, in such a manner that they may readily comprehend it and may constantly be interested. I will not attempt to give any definite method, believing that any teacher with a thorough knowledge of the subject can best evolve his own methods, but will confine myself to some cautions and hints of a general nature, which I have found useful in my own work. In the first place, an emphatic caution against the use of textbooks will be necessary. These would be most pernicious in the hands of young pupils, and would effectually defeat the chief end of the study; for the children would soon learn to rely on the book, and not on their own abilities, and would thus fall into the very habits we wish to avoid. A teacher should find a textbook unnecessary; for a teacher who has not so digested the study, that he or she can develop it without the use of a book, or a teacher who is dependent on the proofs of the book, is not a fit person to undertake teaching Geometry to such a class of young beginners. The first essential, then, for success, is a thorough and philosophical knowledge of Geometry, together with a quick perception of ways and means. A second caution I found necessary, was not to attempt to give the pupils too much, or to advance them too rapidly, though the advance should be sufficient to constantly necessitate some mental effort. Many propositions and definitions are given in the ordinary textbooks, which it would be well to omit at this stage, many applications and extensions not found in the books, it will be well to bring in. We can thus adapt the study to the development of the children, and can teach them the principles involved in a given proposition, without harping on this proposition, which would tend to weaken the interest. It is to be able to illuminate a principle in this manner, that a teacher should have a quick perception of ways and means.
In all geometric work of this nature it will be found most effectual to proceed synthetically. We may start with the idea of points, showing how they may be connected in various ways, and thus come to the idea of lines, when it will be necessary to demonstrate, how they may be divided, how they may vary in length and direction. In all this work it is well to let the children themselves draw the various figures, learning to letter them in the conventional manner. Practical demonstrations will also be of great help, for which the walls of the room, their lines of intersection and corners, or wooden cubes, octohedrons, etc., and natural crystals offer ample opportunity. The various relations and properties to be studied should be discovered by the pupils themselves, with the aid of judicious questions on the part of the teacher, and it is only after they have been discovered and fully comprehended, that their definitions should be given.
From the relations of two straight lines to each other, which would bring in angles and the various propositions relating to them, we would come to the relations of three straight lines and thus reach the triangle.
The above indicates sufficiently, I believe, how the study may be built up and developed. Considerable work will be necessary on the part of the teacher, a careful study of the pupils, and a rational adaptation of the subject to their needs. But the result will justify the trouble, for a conscientious teacher will find himself surrounded by a class of intelligent and bright pupils, ever ready to ask questions, but questions as to the rational why and wherefore of things, children who will have their eyes open and be able to recognize the relations between cause and effect.
BY G. T. JOHNSON.
Oh, the sunlight! the beautiful sunlight!
Dispelling sadness and bringing gladness,
Drifts it in with the spring-time air.
Search forever, yet you will never
Find, of things earthly, aught half so fair!
Softly peeping, silently creeping
Into our eyes, into our hearts.
Oh, the sunlight! the beautiful sunlight!
Leaves us better when it departs.
CIVIL GOVERNMENT IN THE SCHOOLS.
BY FRANCIS NEWTON THORPE, PH. D.
TEARLY a century ago, Madam de Staël said in one of Corinne's letters: 66 The Government of a nation forms its Character." The political experience of the century leads us to reverse the dictum and read it: The character of a nation forms its government. It is our purpose briefly to state our opinion concerning the place of the study of civics in our schools and the goal towards which the study should move.
Civil government in this country belongs to the ancient Teutonic kind, which developed a thousand years in England, and was then transplanted to America. It has been experimental rather than theoretical, and has changed in its civil phenomena from time to time, passing through the phases of feudalism, limited monarchism and representative democracy. To us the first and the second phases are of historical interest; the third divides into two periods, the first, of organization, the second, of administration. The period of organization may be considered, again, in two ways: in a National sense, in the organization of the Federal government, and in a local sense in the organization of the state governments. While civil government in this country was yet in form a limited monarchy, it was constantly encroached upon by an irresistible democracy. The result was a representative democracy. All the colonial governments were of mixed character; in the English national sense, they were of a limited monarchy; in the American local sense, they were of a representative democracy. The revolution as a civil phenomenon was of a national, not of a local type. The colonies became states as it were by merely a vote of the various colonial legislatures, but the government of the United States was an embodiment of the will of the nation. It was formed by the people as a federal unit, and by revolution took the place of the king in Parliament. There was a general concensus of opinion among the members of the Philadelphia Convention of 1787, that a supreme legislative, executive and judiciary should be established. After the adoption of this resolution, the work of the convention was to arrange the details of the Constitution.
In fixing these details the convention followed in general the lessons of civil experience already learned in colonial times. Eleven of the states had already framed constitutions, and in this mine of constitutional organization the convention labored boldly and re-enacted in the constitution for the nation the best provisions found in the constitutions for the states.
The Federal Constitution was not made at a single stroke; it was the result of long civil experience in England and in America. A century of experience under this constitution reveals that those provisions in the Federal constitution which were based upon the test of experience have prescribed civil rights and duties whose observance may be said, generally, to have been recognized without commotion, harmoniously, acceptably, while those provisions based upon theoretical views, such as the provision for presidential electors, have never given satisfaction. The last five amendments to the Constitution of the United States are sufficient to prove that the government is determined by the character of the people. In those amendments may be read in the most solemn form the experience of the people of the United States during the first century of their national history.
These five amendments embody the experience of the American people in the solution of the great civil problems of the century: 1. The inability of a state; resulting in a provision in the Constitution which seriously imperils the legal right to enforce the moral responsibility of a state.
2. The manner of choosing the President and the Vice-President; resulting in a provision which has seated four chief magistrates who did not receive the greatest number of popular votes, and which is a cause of popular irritation every four years. This problem will ultimately be solved anew.
3. Slavery; resulting in its abolition.
4. Citizenship; resulting in a definition of federal citizenship and of state citizenship.
5. The right of suffrage; resulting in the dictum that a state determines the conditions under which a citizen of the United States becomes an elector.
From this brief reference to the Federal Constitution we conclude that national experience determines the character of the national constitution. In other words, the organization of the federal government was based upon colonial and early state expe