On Teaching (1950-1959)

"It has been said, often enough and certainly with good reason, that teaching mathematics affords a unique opportunity to teach demonstrative reasoning. I wish to add that teaching mathematics also affords an excellent opportunity to teach plausible reasoning. A student of mathematics should learn, of course, demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet he should also learn plausible reasoning; this is the kind of reasoning on which his creative work will mainly depend, The general student should get a taste of demonstrative reasoning; he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. He needs, however, in all his endeavors plausible reasoning. At any rate, an ambitious teacher of mathematics should teach both kinds of reasoning to both kinds of students." (George Pólya, "On Plausible Reasoning", Proceedings of the International Congress of Mathematics, 1950)

"Statistics is not the easiest subject to teach, and there are those to whom anything savoring of mathematics is regarded as for ever anathema." (Michael J Moroney, "Facts from Figures", 1951)

"[...] science taught [...] without a sense of history is robbed of those very qualities that make it worth teaching to the student of the humanities and the social sciences." (I. Bernard Cohen, 1952)

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"For that theory [mathematical theory of statistics] is solely concerned with working out the properties of the theoretical models, whereas what matters - and what in one sense is most difficult - is to decide what theoretical model best corresponds to the real world-situation to which statistical methods must be applied. There is a great danger that mathematical pupils will imagine that a knowledge of mathematical statistics alone makes a statistician." (David G Champemowne, "A Discussion on the Teaching of Mathematical Statistics at the University Level", Journal of the Royal Statistical Society Vol. 118, 1955)

"It is proper to the role of the scientist that he not merely find new truth and communicate it to his fellows, but that he teach, that he try to bring the most honest and intelligible account of new knowledge to all who will try to learn." (J Robert Oppenheimer, "The Open Mind", 1955) 

"The true responsibility of a scientist, as we all know, is to the integrity and vigor of his science. And because most scientists, like all men of learning, tend in part also to be teachers, they have a responsibility for the communication of the truths they have found. This is at least a collective, if not an individual responsibility. That we should see in this any insurance that the fruits of science will be used for man’s benefit, or denied to man when they make for his distress or destruction, would be a tragic naiveté."  (J Robert Oppenheimer, "The Open Mind", 1955) 

"At bottom, the society of scientists is more important than their discoveries. What science has to teach us here is not its techniques but its spirit: the irresistible need to explore." (Jacob Bronowski, "Science and Human Values", 1956)

"There are two ways to teach mathematics. One is to take real pains toward creating understanding - visual aids, that sort of thing. The other is the old British style of teaching until you’re blue in the face." (James R Newman, New York Times, 1956)

"Just as there is an applied mathematics of games, genetics, and mechanics, so there should be an applied mathematics (at least in terms of concepts, perhaps with techniques and operations) of the applications of mathematics. When there is, mathematicians will be able to teach the applications of mathematics." (John W Tukey, "The Teaching of Concrete Mathematics", The American Mathematical Monthly Vol. 65 (1), 1958)

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958) 

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"The world of today demands more mathematical knowledge on the part of more people than the world of yesterday and the world of tomorrow will demand even more. It is therefore important that mathematics be taught in a vital and imaginative way which will make students aware that it is a living, growing subject which plays an increasingly important part in the contemporary world." (Edward G Begle, "The School Mathematics Study Group," The Mathematics Teacher 51, 1958)

"We believe that student will come to understand mathematics when his textbook and teacher use unambiguous language and when he is enabled to discover generalizations by himself." (Max Beberman, "An Emerging Program of Secondary School Mathematics", 1958)