On Teaching (1925-1949)

"Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"The primary purposes of the teaching of mathematics should be to develop those powers of understanding und analyzing relations of quantity and of space which are necessary to an insight into and a control over our environment and to an appreciation of the progress of civilization its various aspects, and to develop those habits of thought and of action which will make those powers in effective in the life of the individual." (J W Young [Ed] The Reorganization of Mathematics in Secondary Education, 1927)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Before teachers can properly correlate mathematics with other fields, they ought to learn how to correlate the various parts of mathematics. They should first learn how and where arithmetic and informal geometry can be correlated, how and where algebra can be best correlated with arithmetic and informational geometry, and so on. Unless we can do this, there is small chance that we can successfully correlate mathematics with science, music, the arts, and other applied fields." (W. D. Reeve "Mathematics and the Integrated Program in Secondary Schools", Teachers College Record 36, 1935) 

"Many teachers and textbook writers have never recognized the power of sheer intellectual curiosity as a motive for the highest type of work in mathematics. and as a consequence they have failed to organize and present the work in a manner designed to stimulate the student's interest through a challenge to his curiosity." (Charles H Butler & E Lynwood Wren, "The Teaching of Secondary Mathematics, 1941)

"One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles." (George Pólya, "How to Solve It", 1945)

"The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach." (George Pólya, "How to solve it", 1945) 

"When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should. first of all, stir up his curiosity, give him some desire to solve the problem. The teacher should also allow some time to the student to make up his mind to settle down to his task. Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through success, to appreciate small advance, to wait for the essential idea, to concentrate with all his might when it appears, If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point." (George Pólya, "How to Solve It", 1945) 

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution." (Richard Curand [forward to Carl B Boyer’s "The History of the Calculus and Its Conceptual Development", 1949])