"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Henri Poincaré, "The Value of Science", 1905)
"The aims in teaching geometry should he, according to my views: (1)That the pupil should acquire an accurate thorough knowledge of geometrical truths. 2. That he should develop the power o! original, logical, geometrical reasoning. 3. That he should acquire a habit of thought which will give him a practical sagacity; which will develop his judgment, increase his resourcefulness, and fit him to cope more successfully with the many and varied problems of his after life; which will teach him to rake a many-sided view of things, that if the avenue of attack is blocked, he should able to promptly, cheerfully and successfully attack from another." (W E Bond, "The Aims in Teaching Geometry and HOW to Attain Them", The Mathematics Teacher, 1908) [source]
"There is no science which teaches the harmonies of nature more clearly than mathematics […]." , (William Andrews, "Magic Squares and Cubes", 1908)
"It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction." (Felix Klein, "Lectures on Mathematics", 1911)
"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)
"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"To come very near to a true theory, and to grasp its precise application, are two very different things, as the history of a science teaches us. Everything of importance has been said before by somebody who did not discover it." (Alfred N Whitehead, "The Organization of Thought", 1917)
"Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. […] Let him explain […] science is not a dead system - the excretion of a monstrous pedantism - but really one of the most vigorous and exuberant phases of human life." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)
"No student ought to complete a course in mathematics without the feeling that there must be something in it, without catching a glimpse, however fleeting, of its possibilities, without at least a few moments of pleasure in achievement and insight." (Helen A Merrill, 'Why Students Fail in Mathematics", The Mathematics Teacher, 1918) [source]
"Most teachers waste their time by asking questions which are intended to discover what a pupil does not know whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing." (Albert Einstein, 1920)
"[…] teachers are simply your guides. You yourselves must do the travelling." (William J Gies, "Research in Destiny", 1921)
"Our work is great in the classroom it we feel the nobility of that work, if we love the human souls we work with more than the division of fractions, if we love our subject so much that we make our pupils love it, and if we remember that our duty to the world is to help fix in the minds of our pupils the facts of number that they must have in after life." (David E Smith, "The Progress of Arithmetic", 1923)
"We have come to believe that a pupil in school should feel that he is living his own life naturally. with a minimum of restraint and without tasks that are unduly irksome; that he should find his way through arithmetic largely hoy his own spirit of curiosity; and that he should be directed in arithmetic as he would he directed in any other game, - not harshly driven, hardly even led, but proceeding with the feeling that he is being accompanied and that he is doing his share in finding the way." (David E Smith, "The Progress of Arithmetic", 1923)
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