I'm interested to read writings about mathematics education written by famous mathematicians. By famous mathematicians, I mean roughly anybody with a result or object named after them.
I'm not sure what the format of answers should be, but perhaps one link per answer would be good, so that votes and discussions can be on individual articles, and perhaps each answer could include a short excerpt from the article to avoid having just a long list of links.
On teaching mathematics by V. I. Arnold
In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun).
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The theorem of classification of surfaces is a top-class mathematical achievement, comparable with the discovery of America or X-rays. This is a genuine discovery of mathematical natural science and it is even difficult to say whether the fact itself is more attributable to physics or to mathematics. In its significance for both the applications and the development of correct Weltanschauung it by far surpasses such "achievements" of mathematics as the proof of Fermat's last theorem or the proof of the fact that any sufficiently large whole number can be represented as a sum of three prime numbers.
Mathematical Education by Bill Thurston
The typical response of American adults, on meeting a mathematician, is one of dismay. They apologetically recall the last mathematics course they took, which is usually the one where they lost their grip on the subject matter.
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I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore, while 134/29 was an object with no implicit work. I went excitedly to my father to explain my major discovery. He told me that of course this is so, $a/b$ and $a$ divided by $b$ are just synonyms. To him it was just a small variation in notation. [...] Mathematics is full of this kind of thing, on all levels. It never stops.
Calculus Reform – For the Millions by David Mumford
Watching my children move through the mathematical curriculum of elementary and high schools has been very instructive for me. For instance, I believe there is no universal best way to teach mathematics which applies to all the basic skills. It is very tempting to adopt some pedagogical theory or intellectual standpoint and convince yourself that this is the yellow brick road leading to understanding. I am not convinced that the experts who study pedagogy in mathematics have a deeper insight into what works than most concerned parents.
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A very interesting point relating to the use of formal definitions has been raised by Saunders Mac Lane and other critics of the Gleason-Hallett text. They object to the definition of a continuous function—“the closer $x$ gets to $a$, the closer $f (x)$ gets to $f(a)$” in this book—raising the example of $x \sin{1/x}$. [...] For the sentence above which purports to define continuity, I suggest you ask a mathematically naive friend whether they find anything odd about the sentences: “A clock pendulum is slowing down from friction. As it does so, it gets closer and closer to the vertical position.” Or ask whether the assertion that “runner $x$ is getting closer and closer to a new world’s record in the 100-meter” implies that $x$ never has a bad day? I certainly agree that a footnote clarifying the Gleason-Hallett definition to say that $f(x)$ need not go straight to $f (a)$, but may wobble on the way, is appropriate. But whether the Gleason-Hallett definition is correct as it stands is not a well-posed question: virtually the only sentences in English with an unambiguous interpretation (not requiring the use of common sense by the reader) are those written in mathematical jargon.
