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I suspect that school and university math exams emphasize meticulous and flawless computation over intuition, understanding and problem solving. So to get good

I suspect that school and university math exams emphasize meticulous and flawless computation over intuition, understanding and problem solving. So to get good grades, it's not good enough to understand math, you also have to be able to focus enough to avoid mistakes. Between ADHD and the prefrontal cortex developing/maturing well past the age of 20, that's quite an ask.

msla 7y ago
Estimation should be taught as a respectable discipline and one worthy of serious study. Even aside from the fact no intelligent educator can claim with a straight face that most of their students will need to do arithmetic by hand in a real scenario, estimation is vital for getting an intuition about orders of magnitude and how quickly addition and multiplication increase a value.
philipov 7y ago
My grandfather was a mathematician and he once admonished me that understanding math is as useful as understanding swimming. If you're drowning, you need to be able to DO swimming, not just understand it. Understanding will come from practice, but not the other way around.
- thrwayxyz 7y ago
  Absolutely. People who understand math can't swim. They just board a ship going in the direction they want and get there without drowning.
  - philipov 7y ago
    Yeah, and if they want to go somewhere there isn't already an existing shipping route, they're out of luck! And it took me a long time to realize for myself that with math, the proof of a theorem is actually isomorphic to understanding it.
    - thrwayxyz 7y ago
      Proof by contradiction would disagree with you.
      - philipov 7y ago
        That's a good point, my observation is limited to constructive proofs. Proofs by contradiction or enumeration might not offer any great insight, unless one can prove that there exists no better proof, in which case, the insight is that no great understanding is to be had. Godel's Incompleteness Theorem teaches us that sometimes things are true simply because they are not false. I don't think that contradicts my point, it just limits its scope.
        thrwayxyz 7y ago
        What's constructive in one system is nonconstrictive in another. The only people who take constructive proofs to be more meaningful than nonconstrictive ones are software engineers. Usually with some mumbo jumbo about how type systems are a superior foundation for maths to sets.

msla 7y ago

Estimation should be taught as a respectable discipline and one worthy of serious study. Even aside from the fact no intelligent educator can claim with a straight face that most of their students will need to do arithmetic by hand in a real scenario, estimation is vital for getting an intuition about orders of magnitude and how quickly addition and multiplication increase a value.

philipov 7y ago

My grandfather was a mathematician and he once admonished me that understanding math is as useful as understanding swimming. If you're drowning, you need to be able to DO swimming, not just understand it. Understanding will come from practice, but not the other way around.

thrwayxyz 7y ago
Absolutely. People who understand math can't swim. They just board a ship going in the direction they want and get there without drowning.
- philipov 7y ago
  Yeah, and if they want to go somewhere there isn't already an existing shipping route, they're out of luck! And it took me a long time to realize for myself that with math, the proof of a theorem is actually isomorphic to understanding it.
  - thrwayxyz 7y ago
    Proof by contradiction would disagree with you.
    - philipov 7y ago
      That's a good point, my observation is limited to constructive proofs. Proofs by contradiction or enumeration might not offer any great insight, unless one can prove that there exists no better proof, in which case, the insight is that no great understanding is to be had. Godel's Incompleteness Theorem teaches us that sometimes things are true simply because they are not false. I don't think that contradicts my point, it just limits its scope.
      - thrwayxyz 7y ago
        What's constructive in one system is nonconstrictive in another. The only people who take constructive proofs to be more meaningful than nonconstrictive ones are software engineers. Usually with some mumbo jumbo about how type systems are a superior foundation for maths to sets.