I agree with the author entirely. But I still feel it's important to be able to do computation in your head if just to develop that "number sense". As a physicist, I use computers to do all my real calculations. But when I get the answer that seems wrong, I can mentally do some basic calculations that convince myself the computer-answer I got is probably off by a factor of two, and look for the bug.
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
#12I think that if you want to understand how ordinary people use math, a good place to start would be by looking at their Excel spreadsheets. Excel is arguably the most widely used tool for computation and even programming. I've seen some impressive spreadsheets for solving complex problems, written by folks with no more than a high school education.
With that said, I'd endorse a move away from solely algorithm based (standardized test based) learning.
Re: Most Math Problems Do Not Have a Unique Right Answer
#13As 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'.
No I have read the story elsewhere, the reasoning is that if they were shot any more they wouldn't have returned at all (hence skewing the statistics). So they can take lots of hits on other spots and come back safely, but not so much on the engine so we protect the engine
Re: Most Math Problems Do Not Have a Unique Right Answer
#14I 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…
> I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles
Personally I think this is a myth and the brain is not like some muscle that needs to be kept in shape with a specific set of exercises. Certainly stimulating the brain has been shown to have positive effects in older people, but it hasn't been proven that there's a difference between learning a new language or playing bridge. And nothing proven was found to prevent Alzheimer.
Generally speaking, offloading computations to a machine allows one to concentrate his gray matter on other things and before we come to a conclusion that this leads to brain atrophy, evidence needs to be presented.
Re: Most Math Problems Do Not Have a Unique Right Answer
#15As 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'.
Bullets can be considered almost random in where they will hit, so, if it gets back with lots of holes in the fuselage, it tells you that you can have lots of holes in the fuselage and it still gets back. If almost none return with hits to the engine, then almost every engine hit is bringing down a plane.
Re: Most Math Problems Do Not Have a Unique Right Answer
#16It is refreshing to find an article with a false title that contains it's own refutation within the first five paragraphs:
"The only career in which a high school graduate can expect to continue to work on [problems with a unique correct answer] is academic research in pure mathematics"
Unfortunately for this article, the premise that mathematics is the same thing as engineering is false.
If an engineering problem does not have a unique solution, its because of complications introduced by the real world. Any engineering problem which can be well-posed as a pure math problem, does of course have a unique solution; as the author concedes.
Re: Most Math Problems Do Not Have a Unique Right Answer
#17I 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…
> I hypothesise that offloading this brain work to a machine atrophies your cerebral muscles Personally I think this is a myth and the brain is not like some muscle that needs to be kept in shape with a specific set of exercises. Certainly stimulating the brain has been shown to have positive effects in older people, but it hasn't been proven that there's a difference between learning a new language or playing bridge…
Trying to view the brain as "just a muscle" is probably simultaneously helpful and misleading in different ways.
But I believe the parent's point was more that what is taught is becoming fundamentally different. If you learn how to add using a calculator are you learning to add or learning how to use a calculator? It's a subtle distinction but in practice the ramifications become important.
Re: Most Math Problems Do Not Have a Unique Right Answer
#18I 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…
There is a funny story about Gauss from the book Men of Mathematics, in which Gauss asked two of his students to compute a certain infinite series to 30 decimal places. After many days and nights, the students had a disagreement over what number was in the 30th decimal place, so they asked for help. After looking at the calculations for a few seconds, Gauss accurately pointed at the correct answer and went back to what he was doing.
Re: Most Math Problems Do Not Have a Unique Right Answer
#19It is refreshing to find an article with a false title that contains it's own refutation within the first five paragraphs: "The only career in which a high school graduate can expect to continue to work on [problems with a unique correct answer] is academic research in pure mathematics" Unfortunately for this article, the premise that mathematics is the same thing as engineering is false. If an engineering problem do…
It depends on what you mean by a math problem. It seems a common idea is: A math problem is a question asking for the set of solutions to some mathematical equation, and only the actual solution is useful.
But I would include all of the following as math problems: -Prove that some equation actually has solutions -Find some bounds on solutions to that equation -How can I compute an approximate solution to that equation? -How good is that approximation going to be? -What's a way to try to separate large data sets into clusters?
With regards to the first four: many equations people are interested in simply have no hope of getting a solution you can write down. Simple example: sqrt(2). A more complicated example: solutions to the Navier-Stokes equations.
Re: Most Math Problems Do Not Have a Unique Right Answer
#20And 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…
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