On Teaching (1950-1974)

"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)

"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)

"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)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...]  is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"But both managed to understand mathematics and to make a 'fair' number of contributions to the subject. Rigorous proof is not nearly so important as proving the worth of what we are teaching; and most teachers, instead of being concerned about their failure to be sufficiently rigorous, should really be concerned about their failure to provide a truly intuitive approach.. The general principle, then, is that the rigor should be suited to the mathematical age of the student and not to the age of mathematics." (Morris Kline, "Mathematics: A Cultural Approach", 1962) 

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck "Teaching Machines and Mathematics Programs",  The American Mathematical Monthly 69, 1962) 

"Science is a way to teach how something gets to be known, what is not known, to what extent things are known (for nothing is known absolutely), how to handle doubt and uncertainty, what the rules of evidence are, how to think about things so that judgments can be made, how to distinguish truth from fraud, and from show." (Richard P Feynman, "The Problem of Teaching Physics in Latin America", Engineering and Science, 1963)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, let nothing else be learned than learning. His conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information." (Martin Heidegger, "What is called thinking?", 1968)

"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 & F Lynwood Wren, "The Teaching of Secondary Mathematics" 5th Ed., 1970)