I think this article - "A Mathematician's Lament" by Paul Lockhart - is something everyone should read Not exactly an answer to your questions and not exactly aimed at math beginners, unfortunately, but it's a really engaging rant about this topic https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf …https://twitter.com/graciegcunning/status/1298804338727489536 …
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Replying to @arthur_affect
I appreciate she's asking questions about the topic. I don't understand math at all, never did, and I counted myself lucky to pass math with a C-. I hated it all with the burning intensity of a thousand white-hot suns
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Replying to @suburbanbeatnik @arthur_affect
to me, it was all... blah blah blah blah GINGER blah blah blah blahpic.twitter.com/Ig2iwKz9BL
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Replying to @suburbanbeatnik
Yeah and I dunno on some level that's fine You're a grownup now and you're living your life and you know all the things you need to know to live it, more or less
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Replying to @arthur_affect @suburbanbeatnik
Lockhart's Lament is partly him just getting mad and asking what the point of math class is at all Like the traditionalists say the really important thing isn't all this highfalutin philosophical bullshit, it's whether you can work out the square root of 7 with a pencil if asked
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Replying to @arthur_affect @suburbanbeatnik
Hell, I'm a maths graduate, and I'm not sure I could work that out using the traditional method. (I could do it by an iterative process, mind. "OK, I know 27/10 is a decent approximation, so 70/27 must also be, and their average will be better...")
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Replying to @IMJackRudd @suburbanbeatnik
My mom taught me how to do it as a kid, since she took math classes in an earlier generation when they assumed you'd have to do more things by pencil and paper I've kept it in my back pocket ever since as a party trick
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It's beyond me to explain it in words concisely but it's a long-division-like process based on the algebra for squaring a binomial - (a+b)^2 = a^2 + 2ab + b^2
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So it's like, the square root of 7 First I come up with the "biggest number I can square that fits into 7 without a remainder", just like long division So that's 2
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2 is now the "a" in a binomial expressed as (10a + b)^2, which expands to 100a^2 + 20ab + b^2 Now to continue to the next decimal place of the solution I need to know what b is
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I expand 7 to 7.00 and subtract 4 (this is me subtracting 100a^2 from the expansion) I am left with 3.00, which is 20ab + b^2, or b(20a + b) The correct value for b is one that leaves the smallest remainder from the number I actually have without going over, like long division
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So using the formula (2a + b)*b, I double the digit I just wrote down, 2 I get 4, I make the 4 into 4_ and try to guess a next digit (forty-what?) that I can multiply it by (41*1? 42*2?) so that it "fits into" 300
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The highest digit I can plug into 4_ * _ that "fits into" 300 is 6 -- 46*6 is 276, 47*7 is 329
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