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A B and C D A D E absurdity ADC and C D B adjacent angles angle A E C angles A D E angles ADC Anna apothem arc A B arc C D Archimedes C D B is equal C D is perpendicular called curved surface decreasing variable define definition demonstration difference direction drawing the figure Euclid explained F E C geometrical solid Helen hence Henry Herbert Spencer hypotenuse illustrate incommensurable indefinitely approaching inscribed polygon Julia Leibnitz lesson Let the line line A B line B C lines meeting magnitudes Mary mathematical mean memorizing mental method method of exhaustion perpendicular to B C Plane Geometry principles proof proposition prove pupil ratio reason recitation reductio ad absurdum scholars sides straight line studying geometry suppose supposition teacher teaching theorem theory of limits tion triangle variable approaching
Page 131 - In the first place, it guarantees a vividness and permanency of impression which the usual methods can never produce. Any piece of knowledge which the pupil has himself acquired, any problem which he has himself solved, becomes by virtue of the conquest much more thoroughly his than it could else be.
Page 13 - ... part of his school discipline, but rather the insight into the meaning of processes, and the training in logic. If Algebra and Geometry do not make the student a clearer and more accurate and more consecutive thinker, they are worth nothing. And in the new revolt against the long supremacy of Euclid, as represented in the Syllabus of the " Association for the Improvement of Geometrical Teaching...
Page 136 - To give the net product of inquiry, without the inquiry that leads to it, is found to be both enervating and inefficient. General truths to be of due and permanent use, must be earned. " Easy come easy go," is a saying as applicable to knowledge as to wealth.
Page 128 - ... unfit for useful activity. Others perceive how little this accumulation of abstract knowledge avails in preparation for active life, and direct their attention almost exclusively to matters of a practical nature. On this plan, there is no small danger of producing mere instruments for others — men almost incapable of original thought or independent action.
Page 130 - ... suddenly had their intellects roused by thus ceasing to make them passive recipients, and inducing them to become active discoverers. The discouragement caused by bad teaching having been diminished by a little sympathy, and sufficient perseverance excited to achieve a first success, there arises a revulsion of feeling affecting the whole nature. They no longer find themselves incompetent ; they, too, can do something. And gradually as success follows success, the incubus of despair disappears,...
Page 129 - To bisect a line, to erect a perpendicular, to describe a square, to bisect an angle, to draw a line parallel to a given line, to describe a hexagon, are problems which a, little patience will enable him to find out And from these...
Page 136 - The faculties are called into no exercise by doing a thing merely because others do it, no more than by believing a thing only because others believe it.
Page 130 - It has repeatedly occurred that those who have been stupefied by the ordinary school-drill — by its abstract formulas, its wearisome tasks, its cramming — have suddenly had their intellects roused by thus ceasing to make them passive recipients, and inducing them to become active discoverers. The discouragement caused by bad teaching having been diminished by a little sympathy, and sufficient perseverance excited...
Page 121 - Recall that the area of a rectangle is equal to the product of the measure of its length and the measure of its width.
Page 131 - ... further questions. The solution of yesterday's problem helps the pupil in mastering to-day's. ' Thus the knowledge is turned into faculty as soon as it is taken in, and forthwith aids in the general function of thinking — does not lie merely written on the pages of an internal library, as when rote-learnt.