To teach and learn

When you post your solution of a math problem somewhere, there are, very roughly speaking, 2 ways to present it:

1) Like writing a math paper; proving everything strictly, without appealing (too much) to intuition and usually written in a totally different order, than the way it was solved
2) Describing the whole road you took; starting with the problem, initial thoughts, ideas, fallbacks, vague notions etc.

Clearly something in between is possible too. Anyway, I think it’s quite obvious that I went with 2) when I blogged about the coin-flipping problems. And since way 1) of presenting math has a lot of advantages, you might wonder why I went with 2). I’ll tell you.

If someone else than me is actually trying to gain anything by this blog, chances are that he or she is not used to the way math is usually presented. Because, from my personal experience, when I talk to a non-mathematician about math and am, for example, explaining my proof of some theorem or my solution to a problem, the response I get the most is ‘alright, I get this and that when you explain it, but I would never be able to come up with it myself’. This could obviously be due to the fact that that person is just not as smart as I am. But I beg to differ. Well, of course, it’s usually trivially true that I am smarter than that person, but that’s not the issue; I don’t think that’s the reason. The reason that that person (ah, let’s just make that person a she), the reason that she is not able, or thinks she’s not able, to come up with a solution, is just because she never learned to think like a problem-solver. So, if there’s anyone out there that reads this blog and tries to learn something from it, I want to help her (yeah, it’s definitely a her) as much as I can. And I think the way to teach somebody to solve a problem, is to show her some possible lines of thought. Show her the things you thought when you thought about the problem. Show her the ways you tried to overcome certain obstacles on the way. Show her some problem-solving techniques. Some will work on some problems. Most won’t work on most problems. But that’s ok. Because when you give a non-mathematician a problem and ask what line of attack is the best at first sight, she usually doesn’t have a clue. And I think teaching someone how to get an idea, only a tiny little idea to start with, is all you have to do.

Alright, let me rephrase the above paragraph in a language that’s somewhat more egocentric;

Since I´m probably the only one that is actually trying to gain anything by this blog, I have to realise that I´m not used to the way math is usually presented. Because, from my personal experience, when I read about math and am, for example, trying to get someone´s proof of a theorem, or some guy´s solution to a problem, the thing I think the most is ´alright, I don´t get this, nor that, let alone that I will ever be able to come up with it myself.’ This could obviously be due to the fact that that person is just smarter than I am. But I beg to differ. Well, of course, it´s usually trivially true that that person is smarter than I am, but that´s not the issue; I don’t think that’s the reason. The reason that I am (ah, I wish I was a girl), the reason that I am not able, or think that I’m not able, to come up with a solution, is just because I never really learned how to think like a problem-solver. So, since I’m writing this blog for myself and I am going to try to learn something from it, I should try to help myself (nah, being a boy is just fine) as much as I can. And I think the way to teach myself how to solve a problem, is to write down some possible lines of thought. Show myself the things I thought, wanted to think, or should have thought, when I thought about the problem. Show myself the ways I could have tried to overcome certain obstacles on the way. Show myself some problem-solving techniques. Some will work on some problems. Most won’t work on most problems. But that’s ok. Because when I see a problem and ask myself what line of attack is the best at first sight, I usually don’t have a clue. And I think, teaching myself how to get an idea, only a tiny little idea to start with, is all I have to do.

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