Most Math Problems Do Not Have a Unique Right Answer
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Most Math Problems Do Not Have a Unique Right Answer
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Re: Most Math Problems Do Not Have a Unique Right Answer
#2I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, there's a computation of sqrt(2) to 38 decimal places for no discernible purpose. He probably was merely exercising, just like a musician practices scales. Erdős was able to memorise 10 phone numbers at a time from a glance at a phone book, amidst other calculating feats.
I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles. Exercises left to the reader are really exercises in an almost kinesthetic sense.
The computations don't have to be purely numerical: even a diagram chase in abstract nonsense is a valuable exercise. Computation frequently leads to insight. As children are adding two-digit numbers, they start to notice shortcuts about how to perform the computations. These are their own little private theorems, so to speak. So when Devlin here is talking about "mathematical thinking" and the less importance that performing algorithms has today, I hope we don't forget that just because it's less important, it doesn't mean it's not important at all.
Re: Most Math Problems Do Not Have a Unique Right Answer
#3I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…
Historically that's true. But, I suspect there will eventually be a great mathematician who has also suffers from something like severe dyslexia which makes accurate computation vary difficult yet still makes significant contributions.
Re: Most Math Problems Do Not Have a Unique Right Answer
#4I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…
>Riemann's notes ..., there's a computation of sqrt(2) to 38 decimal places for no discernible purpose.
A while back I read about a mathematician taking an informal survey of other mathematicians. One of the questions was if they knew how to manually calculate square roots. (Apparently, the mathematician was part of an older generation that was taught this procedure in class.) Most mathematicians did not know how and a few responded with commentary such as "I don't need to know it in my work and if I had to, it's a procedure I could look up."
Unfortunately, I don't remember if I read about that in a book or a webpage (and google search doesn't find any hits) so I can't give a cite.
The mathematician doing the survey was including it as part of a larger essay. He was explaining a similar theme to the article's author: tedious manual computation exercises are overemphasized in the typical math education.
Re: Most Math Problems Do Not Have a Unique Right Answer
#5I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…
Historically that's true. But, I suspect there will eventually be a great mathematician who has also suffers from something like severe dyslexia which makes accurate computation vary difficult yet still makes significant contributions.
I had a college professor like that. He was nearly unable to perform arithmetic, and would ask for the answer from the class. I wouldn't say he was dyslexic (I have no idea), but he certainly couldn't perform mental arithmetic. But he was fabulous at math. We'd often ask him questions far outside the realm of the textbook and class just for the privilege of seeing an extremely good mind tackle a problem.
So I disagree with the OP. The fact that some mathematicians brains work that way is not evidence that they all do, nor is it evidence that it is helpful or preferable.
Re: Most Math Problems Do Not Have a Unique Right Answer
#6I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…
Historically that's true. But, I suspect there will eventually be a great mathematician who has also suffers from something like severe dyslexia which makes accurate computation vary difficult yet still makes significant contributions.
Mathematicians I knew tended to make minor arithmetical errors as much as anyone else, some of them a lot. Ability to do accurate computation is not much relevant in theoretical math. What is relevant is your ability to reason and abstract thinking.
Of course, that does not mean kids should not be taught arithmetic. It just mean that we should distinguish between mistakes caused by not understanding the topic and those that are just result of doing a lot of operations with similar numbers. the former should be dealt with as bigger deal then latter.
Re: Most Math Problems Do Not Have a Unique Right Answer
#7And yet the way math is traditionally taught in schools and colleges, there is almost always is a unique right answer.
There are alternative approaches to teaching math such as problem-based learning, active learning, and inquiry-based learning, that help students not only understand the math better, but learn why the math is useful and how to apply it in the real-world (transfer of learning). Students also are much more likely to continue on and succeed in future courses and graduate.
Here are best practices for teaching calculus, for example: http://launchings.blogspot.com/2014/01/maa-calculus-study-se...
And research showing that in traditional lecture courses (vs. active learning courses), students are 1.5 times more likely to fail: http://news.sciencemag.org/education/2014/05/lectures-arent-...
Underrepresented populations (minorities) and females are much more likely to succeed in active and inquiry learning math courses, too: http://theconversation.com/who-learns-in-maths-classes-depen...
Re: Most Math Problems Do Not Have a Unique Right Answer
#8As for the armor plating question, did the plate the engine because of its size? The engines combined make up a smaller surface area than the other places but it was still shot up quite a bit. It would be less weight and not to mention it would be more 'mission critical'.
Re: Most Math Problems Do Not Have a Unique Right Answer
#9I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…
Historically that's true. But, I suspect there will eventually be a great mathematician who has also suffers from something like severe dyslexia which makes accurate computation vary difficult yet still makes significant contributions.
My undergraduate mentor was this way. World-class mathematician with dyslexia.
Re: Most Math Problems Do Not Have a Unique Right Answer
#10As for the armor plating question, did the plate the engine because of its size? The engines combined make up a smaller surface area than the other places but it was still shot up quite a bit. It would be less weight and not to mention it would be more 'mission critical'.
my guess is that "evidence" (in the form of bullet holes) shows hits on the engines are the least survivable.
(I am only saying that because I was told that the intuitive answer of "armor plate the most struck parts" is wrong.)