On Teaching (1980-1989)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

“That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects (at a later stage, of marks on paper or even mental images). As a result of this experience, there is something (a trace, an effect) in the pupil's mind which is his representation of the integers. But his representation is equivalent to mine, in the sense that we both get die same answer to any question you ask - or if we get different answers, we can compare notes and figure out what's right. We do this, not because we have been taught a set of algebraic rules, but because our mental pictures match each other.” (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1981)

"I am a firm believer that in studying mathematics one should never forget one’s common sense. Many years ago, I was teaching an elementary algebra course. On one exam, I had a standard-type question that involved finding the ages of the mother, father, and child. After the students read the question, I said, 'On this problem, I’ll give you one hint.' All eyes eagerly turned to me. I continued,: 'If the child should turn out to be older than either of the parents, then you’ve done something wrong.'" (Raymond Smullyan, "5000 B.C. and Other Philosophical Fantasies", 1983)

"It is the very strangeness of nature that makes science engrossing. That ought to be at the center of science teaching. There are more than seven-times-seven types of ambiguity in science, awaiting analysis." (Lewis Thomas, "Late Night Thoughts on Listening to Mahler’s Ninth Symphony", 1983

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"A teacher who is not always thinking about solving problems - ones he does not know the answer to - is psychologically simply not prepared to teach problem solving to his students." (Paul R Halmos, "I Want to Be A Mathematician", 1985)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"More fundamentally students should be taught that instead of asking ‘What techniques shall I use here?,’ they should ask ‘How can I summarize and understand the main features of this set of data?’" (Christopher Chatfield, "The initial examination of data", Journal of the Royal Statistical Society, Series A 14, 1985)

"Probability and statistics are now so obviously necessary tools for understanding many diverse things that we must not ignore them even for the average student." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"The result is that non-statisticians tend to place undue reliance on single ‘cookbook’ techniques, and it has for example become impossible to get results published in some medical, psychological and biological journals without reporting significance values even if of doubtful validity. It is sad that students may actually be more confused and less numerate at the end of a ‘service course’ than they were at the beginning, and more likely to overlook a descriptive approach in favor of some inferential method which may be inappropriate or incorrectly executed." (Christopher Chatfield, "The initial examination of data", Journal of the Royal Statistical Society, Series A 14, 1985)

"Thus statistics should generally be taught more as a practical subject with analyses of real data. Of course some theory and an appropriate range of statistical tools need to be learnt, but students should be taught that Statistics is much more than a collection of standard prescriptions." (Christopher Chatfield, "The Initial Examination of Data", Journal of the Royal Statistical Society A Vol. 148, 1985)

"Experience without theory teaches nothing." (William E Deming, "Out of the Crisis", 1986)

"It is our responsibility as scientists, knowing the great progress which comes from a satisfactory philosophy of ignorance, the great progress which is the fruit of freedom of thought, to proclaim the value of this freedom; to teach how doubt is not to be feared but welcomed and discussed; and to demand this freedom as our duty to all coming generations." (Richard P (Feynman, "What Do You Care What Other People Think?", 1988)

"Science must be taught well, if a student is to understand the coming decades he must live through."(Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)