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Some thoughts people have shared with me or I have learned/read/etc...

- "Math is not a spectator sport."
- Math is motivated by big kid problems, so teaching little kids these things generally fall on deaf ears.
- The historical motivation is not necessarily the best motivation.
- "Drawing is more important than writing set topology arguments tbh."
- Proving is convincing and explaining. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding, not to meet abstract standards of "rigor" or "honesty." (Reuben Hersh)
- "The role of proof in the classroom is different from its role in research. In research its role is to convince. In the classroom, convincing is no problem. Students are all too easily convinced" (Reuben Hersh)
- "A mathematician is young as long as he reads works other than his own." (unknown author, communicated in the preface to Arnold & Khesin's hydrodynamics book)
- There is no barrier to doing mathematics.
- "Most people, traumatized by school experiences of mathematics, know all too well that mathematics is the most meticulous and demanding of disciplines, but few get to see that it is also the most liberating and imaginative of all human activities. Absolute precision buys the freedom to dream meaningfully." (Donal O'Shea)