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Xyven posted:You're probably thinking of A Mathematician's Lament. There are some problems with the article, particularly in how it over-romanticises the subject, but the overall point it makes is good. As someone who did math competitions throughout childhood, graduated with a degree in mathematics, thought for a long time about being a high school math teacher before abandoning the dream for a better paying career, I can really identify with the article. Mathematics done right is absolutely beautiful. Case in point: Rocko Bonaparte posted:I chalked my problems with math to a general lack of discipline since in a 5-step calculation I'll inevitably bungle it completely, but I'm starting to wonder if I would have benefited from another method. Like, I suspect for other folks that alarm bells start going off when the fuckup arrives, not at the end of two pages of hand calculations when the answer doesn't seem to pan out. That was a major problem for me in college. I'd set everything up correctly, get in a few steps, gently caress up, and follow everything to its illogical conclusion. It only came to me a decade later that, you know, that equal sign means I should be able to smash some numbers in at any time and get the same thing on either side of an equation. I felt like an idiot. There's tons of times where I would be in the middle of a derivation and start turning down a wrong path, but I would stop. Once things started to get complicated, they got ugly, for lack of a better word. With practice, you can absolutely pick out the moment where you've messed up instead of "following everything to its illogical conclusion." It's the same sense you get when you hear an off key note or start a faltering sentence. But you can't build that sense from rote math problems/teaching, because they're usually far too short. You have to be left to flounder around for a page or two so that you can start to build that sense of when things aren't working out. Alternatively, you have to learn to sketch out the major derivation steps in your head that you instinctively sense to be true, and then work out a path from there. You might have encountered the term "lemma" in geometry referring to such bridging steps. Again, without "freeform" derivation/proofs, you'll never build that skill. The thing is that beyond a point, math absolutely gets fun. This was a problem I just thought of yesterday, for which I don't know the answer: You know primes. Sometimes primes come in pairs only two apart, like 11 and 13, or 29 and 31. It's been proven that there are an infinite number of such "twin prime" pairs. But what about primes appearing three apart? Or four apart? Some number n apart? Are there an infinite number of "n prime" pairs for any n? If not, which n's? This is a fun question. The answer isn't immediately obvious to me, although I would suspect it's true for all n based on beauty alone. But that's an example of a fun "adult-level" math question that without a proper comprehensive math education, kids would never be able to experience. fake edit: a bit of googling shows that I was wrong, the number of twin primes is only conjectured to be infinite, but there's strong suspicion that it is. My specific question is called Polignac's conjecture, and was first asked in 1849. Thus far, the best efforts of mathematicians have shown that it is true for at least one number N for N < 246. It remains unanswered. Cool! Cantorsdust fucked around with this message at 05:42 on Jul 26, 2014 |
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Jul 26, 2014 05:37
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Xyven posted:It's the same reason it's a problem to over-romanticize anything, it obscures the truth and replaces facts with dreams. The fact is, for most people math is just going to be a tool they use, and pretending that it's not is just naive. The article draws comparisons between art and math, and while there is an element of expression in proofs, proving theorems isn't what math is used for by the vast majority of the population. The analogy to art is fallacious because art is not the fundamental backbone to all the science and technology that drives the modern world. You cannot treat math the same as art, because while someone who does not understand or appreciate art is only missing out on some aspects of culture, someone who does not understand math cannot fully function in society. The thing is, math isn't needed as just a tool to use for the vast majority of people. We have calculators, computers, google, etc to fill the role of that. There's no reason to make basic arithmetic the core focus of a math curriculum. It would be like saying "the vast majority of people use art for making sketches to explain something in a lecture or presentation, so let's remake the art curriculum to focus on mastering the quick sketch." Also, with math, the two goals of high level concepts and low level arithmetic are not mutually exclusive. The low level arithmetic is exclusively derived from the high level concepts, and starting with the concepts first can help to explain the arithmetic. Adding is just iterated counting. Multiplication is just iterated adding. Exponents are just iterated multiplication. And more exotic operations can be described iterating on that, etc. A more concrete example: you probably memorized a bunch of equations for distance/speed/acceleration in your high school physics class. Before calculus, these equations might have seemed arbitrary, but once you learned calculus, you saw the connection between them: speed is the derivative of distance, acceleration is the derivative of speed, jerk the derivative of acceleration, etc. And suddenly, you didn't even need to memorize the equations anymore! You now understood why they were what they were, and you could rederive them if necessary. And loving lol at advanced math being 99% manipulating equations and keeping track of fiddly little details and just 1% being the neat trick that makes the proof. On paper, maybe, but that 1% neat trick is the core of the proof and will take up the majority of the time to come up with. The equations are just writing down the argument you're making in your head. I can't tell you the number of times where I would make some sign error or something, get halfway through the derivation, and realize the sign is opposite what I want. But instead of accepting the arithmetic blindly, my sense of mathematics/aesthetics would tell me to backup and find the mistake somewhere, because what I've derived has to be right, it's just the arithmetic that's wrong. This statement is as absurd as saying that writing a story is 99% using a pen to make words and 1% coming up with characters and a plot. Anyone who's done high level math would tell you otherwise. Lockheart is telling non-mathematicians that those aspects can be ignored almost entirely because most mathematicians ignore those aspects almost entirely. Seriously, university math professors are hilariously forgetful and error-prone sometimes. That's why poo poo like Matlab / Mathematica / Wolfram Alpha exist. Let computers handle the bookkeeping, as they should. You can argue, "but what about the engineers and scientists who are using math instead of playing with it?" Well, that's what computers are for. And before computers, that's what sine tables and slide rules were for. I cannot think of a single example in all of history where a scientist, engineer, or mathematician became known for their amazing skill in arithmetic. But I can remember Gauss as a schoolboy discovering that the sum of a series of consecutive integers from 1 to n equals n(n+1)/2. There's a reason people automated the process of arithmetic and focused on the high level stuff instead. Arithmetic is perfect for a computer: it requires perfect accuracy but no thought. So why are we making children focus on arithmetic instead of math? edit: I guess what I'm arguing is that math is fundamentally closer to being an art or a language as it is a science. It needs to be taught like an art or a language. Cantorsdust fucked around with this message at 17:48 on Jul 26, 2014 |
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Jul 26, 2014 17:42
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Lightanchor posted:There is only one pair of primes appearing three apart. You are correct, I should have revised that to primes some 2n apart.
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Jul 29, 2014 01:23
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