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Mathematics software/advanced calculators and the learning of mathematics.

  • Thread starter Mr.Watson
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  • #26
MarneMath
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It appears the problem for most people is that they believe you want people who want to be professional scientist to go through those classes, where what a future scientist needs is a solid foundation in the 'hard stuff' where computers are not so helpful.

If you envision younger high school age kids being exposed to some more complex ideas and these ideas demostrated via computers, then that's a bit less extremely than what they seem to think you mean.
 
  • #27
symbolipoint
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The answer is both. You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist. I should qualify that, though. There are some things computers can't do, and some things humans can't do (in one lifetime). The question is, do you know which is which?



Hmm. So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing? I hope that anyone with a real passion to understand the world doesn't settle for this.
Sankaku seems to have the best thoughts on the topic. The discussion has otherwise become a mess.
 
  • #28
lurflurf
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The usefulness of computers in algebra, topology, analysis, GR, QFT, classical mechanics, and electromagnetism among others is beyond obvious. Besides being ridiculously error prone and slow hand calculations do not even encompass the same methods as computers use superior methods that are not practical to perform by hand. Sankaku thoughts are particularly contradictory, even though Sankaku acknowledged the existence of problems computers can solve that humans cannot, using them makes one a clueless half scientist drone without understanding. Apparently computing square roots by hand is more worthwhile than computational topology.
 
  • #29
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The problem with learning mathematics using software, is computers will always do what you ask, which may not always be what you want. If you don't understand the math, how can you be sure you are asking the computer the solve the right thing?
 
  • #30
lurflurf
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So your pen and paper does what you want? It is a common concern that a fool with a computer might make an error, as if there is no risk of mistake in hand calculation. Also that the computer user will not understand what they are doing and by taking it away she will know exactly what to do. Lets assume our computer user knows exactly what to do and has spent a thousand hour gaining a deep understanding of the relevant subject and ten minutes doing hundreds of pages of preliminary calculations correctly with the help of the computer. Our pen and paper user has spend the thousand hours doing the same preliminary calculations by inferior and less accurate methods while making 4378 mistakes and ten minutes attempting to learn the relevant subject. In this realistic scenario and with other things being equal (prior to the year in isolation performing the thousand hours and ten minutes of preparation the two were clones with the exact same genetics, knowledge, and experiences) who is more likely to correctly solve the problem first?
 
  • #31
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From an educational standpoint it would make sense to do it both the old fashioned way and with computers. This way you are double exposed to everything and you will hopefully understand it better.

Why should one way be preferred over the other? If the goal is education, you should be learning as much as possible not the bare minimum.
 
  • #32
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Sankaku thoughts are particularly contradictory, even though Sankaku acknowledged the existence of problems computers can solve that humans cannot, using them makes one a clueless half scientist drone without understanding. Apparently computing square roots by hand is more worthwhile than computational topology.
Please, could you explain this statement? I cannot tell what you are criticizing here.
 
  • #33
lurflurf
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^You say

You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist.
So a scientist who cannot do their work without a computer (most of them) is a half scientist which I take it is bad. However many in this thread would praise a technophobe scientist who never uses a computer which you would not you go on to say
There are some things computers can't do, and some things humans can't do (in one lifetime).
Which seems to be in conflict with the previous statement as we must know how to do by hand that which can only be done by computer. Finally
So the world needs a bunch of clueless drones (who do not actually understanding the calculations behind mathematics for 90% of the task in their disciplines) who can punch numbers into a screen but have no idea what they are actually doing?
Yes the world needs such people they can do their work perfectly well without understanding how their roots, linear algebra, Fourier transforms, multiplication, statistical analysis, group theory, or whatever else are actually calculated. The experts in those fields know. Like wise they do not understand most of their own field much less any related field they might use results from. This is also not a problem. This is why we have specialists, no one person can understand everything.
 
  • #34
lurflurf
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From an educational standpoint it would make sense to do it both the old fashioned way and with computers. This way you are double exposed to everything and you will hopefully understand it better.

Why should one way be preferred over the other? If the goal is education, you should be learning as much as possible not the bare minimum.
It is about learning the best tool for the job. The reason learning useless hand calculation is not better than not is because in the time one spends learning them cannot be used to learn something useful. Pilot do not spend a portion of their training trying to fly with there arms and carpenters do not practice pounding in nails by hand. I suspect most people here agree and do not themselves do square roots and trigonometry by hand, but they think doing integrals and algebra by hand is important.
Until you can write a program that can solve all the excercises in a graduate differential topology, analysis, GR, QFT, classical mechanics text. etc., your argument has no real ground.
This is wrong in so many ways.
-All? You would not be impressed by a program that solved 99% of the exercises?
-Are all these exercises solvable by humans? Many books intentionally or unintentionally include very difficult or impossible exercises.
-Does the program need work alone? What about a program that is very helpful to a human in doing the exercises?
-Probably the exercises you are thinking of are meant to be done by humans. What about a books whose exercises are meant to be done with a computer? You must realize that by limiting exercises to those that can be done by the average reader without a computer in a short time greatly limits the learning. All those books have simple contrived problems. Consider a spherical cow.
 
  • #35
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^You say

You should be able to do it by hand and you should be able to use a machine. To be able to do only one makes you half a scientist.

So a scientist who cannot do their work without a computer (most of them) is a half scientist which I take it is bad.
This is a straw-man argument. I was trying to make the point that it is not an either/or situation. This thread was veering off toward "pencil vs machine" arguments.

However many in this thread would praise a technophobe scientist who never uses a computer which you would not you go on to say

There are some things computers can't do, and some things humans can't do (in one lifetime).

Which seems to be in conflict with the previous statement as we must know how to do by hand that which can only be done by computer.
We must know how to do it by hand, yes. This doesn't mean that all the work needs to be done by hand. It means you understand the theory of what you are doing on the computer.

Finally

So the world needs a bunch of clueless drones who can punch numbers into a screen but have no idea what they are actually doing?

Yes the world needs such people they can do their work perfectly well without understanding how their roots, linear algebra, Fourier transforms, multiplication, statistical analysis, group theory, or whatever else are actually calculated. The experts in those fields know. Like wise they do not understand most of their own field much less any related field they might use results from. This is also not a problem. This is why we have specialists, no one person can understand everything.
That is fine, if your aspirations are to be a semi-competent technician. I was assuming that people here wish to become scientists who seek to understand the world. Of course, as you say, no one person can understand everything. However, if you are interested in discovery, you should be more curious about how the tools you use actually work.

Let me ask you for your recommendations for mathematical software and what parts of mathematics you find them particularly useful for. I am happy to tell you what I like, but I think you can go first as the advocate of machine solutions to "99% of the exercises" in my differential topology textbook.
 
  • #36
lurflurf
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The whole thread is a pencil vs machine argument. What do you mean by "understand the theory of what you are doing on the computer. " You object to those "who do not actually understanding the calculations behind mathematics". If a computer tells me
sin(1)~0.84147098480789650665250232163029899962256306079837106567275170999
I certainly do not actually understanding the calculations. I also do not know how to do it by hand. I cannot imagine that knowing how would be helpful. I think doing lots of tedious useless hand calculations will result in less discovery.
Many software packages are helpful for different things
Latex
For typsetting
mathematica
For integrals and such
Magma
for algebra
Matlab
for something sometime
Mathics
http://www.mathics.net/ [Broken]
Which is quite basic
Sage
which include many helpful packages including
R
for statistics
PARI/GP
for arithmetic
LinBox
for linear algebra
Maxima
for symbolics
 
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  • #37
MarneMath
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You really don't know how to do sin(1) by hand? That's a bit odd. Interestingly enough, knowing how a calculator obtains answer is helpful knowledge if for no other reason than a roundoff error. It was a simple roundoff error that allowed the patriot system to fail and get 28 people killed. I'll hate to think some engineer thought to himself, "round off errors! Who cares abou that!"

I think much of Sankaku is a simple, you be aware of the limitation of the technology you us. Graphs can be misleading, computers are known to have bugs, and sometimes certain tools shouldn't be used for certain problems. To be able to figure out work around for these problems, sometimes it requires a good grasp on theory or the 'old fashion way' to solve a problem to see if the computer is giving you a reasonable solution.
 
  • #38
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One of main reasons you need to know how to do it by hand, is someone has to write the stuff that performs the calculations. A computer can do the work, but whose making the computer work? And making sure its doing it right?

Also, floating point computer calculations come with some error due to floating-point arithmetic.

For example a computer cannot contain the exact value of PI in purely a floating point value. This goes for all irrational numbers.

A computer also does not understand what is meant by infinity. Since it is a finite machine. So all applications in physics that use the concept of infinity in their design will not be able to be represented in a machine. This results in further error.

Minor errors might not matter much to some people but if you scale your calculations to something larger, it now has a much larger impact.

There is also the problem that many of the theorems that are from mathematics and physics, are made through proofs. How would a computer prove something when it has no understanding of what is meant by basic language.
 
  • #39
lurflurf
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No one is arguing round off errors should be very large, that is ridiculous. I know how to compute five or ten digits of sin(1), but not hundreds. No it is not the same thing. Even if I was world champion at computing sine by hand a computer would easily beat me. Doing four by four problems in a high school algebra book is not the same as doing million by million problems. Again with the old someone did a calculation on a computer one time and made an error. The person still made the error not the computer. People make errors in hand calculations all the time. If a calculation is impossible for a human that is an accuracy of 0%. In fact the patriot system was the result of a huge blunder, not any inherent problems with computers. That is the lesson, don't make huge blunders. The patriot system did not kill anyone, a scud did. The patriot failed to save them, which is not the same thing. The patriot system should have been able to save them, but it would be impossible without computers. A pad of paper and pen would be entirely useless, that is technology with limitations worth being aware of.
 
  • #40
MarneMath
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I think you're missing the point entirely in some vague attempt to standfast by your position. I don't think anyone here would argue you should be able to do ALL problems without a computer or that a computer cannot help you solve a lot of problems. This isn't a computer all the way or by hand all the way argument. In fact, there shouldn't be argument. You should know how things work. You should be aware that computer programs are designed by people and you should aware that oversights can occur and be able to see if the answer is reasonable. You should be aware of inherent limits of programs.

You seem to believe that people here think you should just do everything by hand. That's not the case at all, the argument is more along the lines that you should know it can be done by hand to some extent and be aware of the technique if for no other reason than to check reasonability. Is there any harm in this? Does learning a simple technique really eat up so much of your time that you rather just be ignorant of it?

If so, then you can carry on your merry well and just believe whatever answer a computer gives if you believe that's all you need to be successful. I, as someone who does most if not all my works on computers (a lot of variable order makov models), would like to be aware of possible errors in the programming and limits of it. But, that might just be me and my need to give a better answer to my boss than simply "uh this is what the computer gave me."

*As a side note about the scud, you can get bogged down in terms and phrases if you wish. I don't care to play that game. Clearly the computer didnt make an 'error' but blind trust in the machine allowed it to fail. The fact of the matter is that it was a simple error. Something that could've and should've been stopped earlier. The computer did what it was programmed to do, and sadly it was programmed poorly. Thus, this only supports my points, you should be aware of how a program works and be aware of possible problems.
 
  • #42
lurflurf
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π=1 (base π)
A pad of paper cannot contain the exact value of π in purely a floating point value (base 2,7,10...). Leaving aside the details of translation of hand calculations into computer algorithms the same methods are not used. Knowing how to calculate something by hand and how to calculate it with a computer are entirely different things. A computer is a useful tool.
 
  • #43
MarneMath
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Is anyone saying that a computer isn't useful?
 
  • #44
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The whole thread is a pencil vs machine argument.
No. All your posts are pencil vs machine arguments. The rest of us are trying to point to an appropriate balance between the two. I think I am done here.
 
  • #45
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I used to worry about whether I was was investing too much time in learning how to do math by hand.

Until I took a Mathematical Computing class.

There were plenty of kids in that class who had VASTLY more experience in programming than me. Most of them had already done projects in Mathematica and just wanted to solidify their knowledge of its ins and outs. I knew next to nothing about programming. But the teacher is notoriously difficult, and he gave us problems that no one managed to do on the first try. Everyone's programs bugged up and did nonsensical things until you dug into them good.

And that's when I discovered that, although I had little experience programming, all my time with pencil and paper math had prepared me better for debugging than anyone in the room. Since I wasn't as comfortable with code, I had Mathematica plot and animate all sorts of visuals representing intermediate steps. I got comfortable creating visuals early and used them to debug rather than searching through lines of code. I took derivatives by hand and then plotted them to check how many roots I SHOULD be getting, and which ones were missing. At one point we were dealing with a function that behaved so badly that FindRoot couldn't keep up with it as parameters changed, so I had to construct a fairly intricate "guess" function that approximately tracked the behavior of the function while being much simpler computationally. I expanded expressions into Taylor series by hand to see what the computer should be telling me in certain extreme cases.

This may sound unsophisticated to people who know all sorts of impressively accurate numerical techniques, but I produced working programs faster than anyone else who I saw working in the lab, and I spent half my time doing things by hand in my notebook. It was mostly due to it being the only thing I felt comfortable with, but it seemed to work really well.
 

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