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Stanos posted:It's really sad to see teachers like Takahashi because it's the EXACT type of people I want teaching but the focus on tests and other problems stifle them. I'm completely terrible at advanced math and the subject is incredibly dry to me and sometimes I wonder if it was just from the rote way it was taught compared to my History and English classes that were far more engaging and interesting. I remember reading an article that I can't find now that goes into 'of course math is loving boring, it's taught in the most boring way possible instead of teaching it like an art form' and I really agreed with it. 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.
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Jul 25, 2014 18:43
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Bigup DJ posted:Oh yeah I agree completely! I'm asking what the problem is with it "over-romanticising" the subject, and more importantly how he came to that conclusion in the first place. He was using it as a pejorative and I'm curious why. 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. Another problem with romanticising math is that, frankly, a lot of it is drudgery. It is EXTREMELY important to keep track of all the little details when you are doing a proof or deriving an equation, and a missed term or forgotten minus sign can completely change your result. In many advanced subjects of math, such as analysis, differential equations, probability/measure theory, your work is 99% manipulating equations and keeping track of fiddly little details and 1% finding a neat trick that makes a proof work. Right now our education system basically focuses solely on those little details, but Lockheart's article leaves non-mathematicians with the impression that they should be ignored almost entirely. In reality these details should be a large part of math education, although not to the exclusion of all else. And again, I do not disagree with the primary point of the article, I just think that he presents an unrealistic picture of a subject that is often dull and tedious.
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Jul 26, 2014 16:18
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Cantorsdust posted: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." Understanding that 1/4 < 1/3 is using math as a tool, understanding how to estimate a 15% tip is using math as a tool, looking at a result that your excel spreadsheet spits out at work and realizing it is off by several orders of magnitude is using math as a tool. There are things that you need to be able to do without a calculator in order to function. And again, the analogy between math and art is terrible, because while art is neat, it is not the underlying mechanic for all of modern science and technology. Cantorsdust posted: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. Writing IS 99% sitting down and putting words on paper, making sure they flow, making sure they all fit. You appeal to your sense of mathematics to recognize when something is wrong without acknowledging where that intuition came from. It comes from a strong understanding of arithmetic and algebraic manipulation that you built up over years of paying close attention to details while doing problems. Cantorsdust posted: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? If you do not have a strong understanding of what the computer is doing, you will be unable to recognize when it spits out clearly nonsensical results. There are also countless examples of times where sitting down and using a computer to do a simple calculation or double check something is a ludicrous waste of time. Nobody is remembered for their amazing skill in arithmetic because it is a prerequisite for further accomplishments. Neither are authors remembered for their great spelling abilities. It's just that you need a solid foundation in order to build up more advanced ideas. Understanding arithmetic is like knowing how to spell; it is something you need to know in order to be a functional human being. I'm sure that math as seen from the eyes of a research mathematician really closely resembles art, but this is the minority view of a fraction of a percentage of the population. Chemists could also describe what they do as close to an art, but it doesn't matter because for most people chemistry is a tool. Math is similar to art in some ways, but that does not mean that math is an art, any more than two sets containing the same object implies that they are both the same set. Let me reiterate that I am not saying there is no room for creativity and expression in math, or that it should be completely ignored. I am saying that Lockheart presents an unrealistic view of math because he has the viewpoint of a research mathematician.
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Jul 26, 2014 18:35
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