A Quote by Max Dehn
Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
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Tarrasch's 'dogmas' are not eternal truisms, but merely instructional material presented in an accessible and witty form, those necessary rudiments from which one can begin to grasp the secrets of chess.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
There is no thing as a man who does not create mathematics and yet is a fine mathematics teacher. Textbooks, course material-these do not approach in importance the communication of what mathematics is really about, of where it is going, and of where it currently stands with respect to the specific branch of it being taught. What really matters is the communication of the spirit of mathematics. It is a spirit that is active rather than contemplative-a spirit of disciplined search for adventures of the intellect. Only as adventurer can really tell of adventures.
It's very eclectic, the way one chooses subjects in the movie business, especially in the commercial movie business. You need to develop material yourself or material is presented to you as an assignment to direct.
The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method.
To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer.
The only way to learn mathematics is to do mathematics.
My main concern with the condition of mathematics in high school is that there's a lot of fear involved! Math is not, generally speaking, presented in a fun way. The concepts, as I see them, are fun, and that's the way I'd like to convey them myself.
On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things whlch are of no concern at all to man.
People tend to think of gentrification in terms of race because it's presented that way, and I think it's presented that way because in poor cities that's what's really going on. Beyond that, I think it's presented that way as a way for the people who are really pushing it to make it just a black problem, so people don't care.
What I think is important about essayists - about the essay, as opposed to a lot of personal writing that kind of finds its way into public view - is that the material really has to be presented in a processed way.
Mathematics never reveals man to the degree, never expresses him in the way, that any other field of human endeavour does: the extent of the negation of man's corporeal self that mathematics achieves cannot be compared with anything. Whoever is interested in this subject I refer to my articles. Here I will say only that the world injected its patterns into human language at the very inception of that language; mathematics sleeps in every utterance, and can only be discovered, never invented.
Mathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability in today's world.
In the Pythagorean system, thinking about numbers, or doing mathematics, was an inherently masculine task. Mathematics was associated with the gods, and with transcendence from the material world; women, by their nature, were supposedly rooted in this latter, baser realm.
Let me introduce the word 'hypertext' to mean a body of written or pictorial material interconnected in such a complex way that it could not conveniently be presented or represented on paper.
In fact, the answer to the question "What is mathematics?" has changed several times during the course of history... It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns.
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