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I complete disagree. Dr.Neal Koblitz (inventor of elliptical curve cryptography)spent a good bit of his life fighting this obsession with bringing the computer

by dxbydt 10y ago

I complete disagree. Dr.Neal Koblitz (inventor of elliptical curve cryptography)spent a good bit of his life fighting this obsession with bringing the computer into math so things can be visualized & manipulated. You should check out what he has to say. https://www.math.washington.edu/~koblitz/mi.html https://www.math.washington.edu/~koblitz/mi.html btw, I used to think like you before I got into math. Its a very valid albeit flawed viewpoint - to think that hey if only there was a javascript doodad somewhere so I could move a slider and manipulate the parameter space, I will instantly "feel" CLT in my blood instead of working through CLT as a dry formal math proof. The reality is, this sort of dumbing things down works maybe in 2D and best case 3D, but beyond that, its utility is rapidly diminished. Also, the results you get in the lower dimensions don't map into higher space. As yummyfajitas points out elsewhere, in a reasonably high dim space, all pairs of random points will be fairly equidistant, which is simply not the case for say 2D. So even the little bit of useful intuition you learn in the lower dims becomes quite useless as you scale up. Proofs are best learnt by doing proofs. Math is not cinema.

dunkelheit 10y ago
Traditional math education: a professor explains proofs of selected theorems on a blackboard and students are then required to memorize the proofs and reconstruct them during the examination. They are also required to solve a lot of problems that involve proving some consequences of the main theorems or doing some rote calculations using nothing but pencil and paper. The advantage of this approach is that it works. You can acquire solid intuition this way. But it is arduous as hell and also pretty one-sided - problems are limited to what an average undergraduate can do by hand. Now, when I think about using computers in math education I don't think of some "javascript doodad" with moving sliders. I imagine instead implementing an algorithm or doing a numerical experiment, learning about possible pitfalls in the implementation, running the algorithm on some real data, seeing it fail etc. I think this experience is pretty valuable too and can nicely complement traditional approach.
- davidivadavid 10y ago
  That's why that approach has been offered in courses traditionally called "Numerical analysis" that have been available to math students for... how long?
  - dunkelheit 10y ago
    Taking the sarcasm out, your point is that such thing exists. Agreed. But usually it is just a single course, it does not cover everything. And why it is a separate course at all? Why should you study some field with pencil and paper and then (possibly, after a couple of years) study a subset of that field "numerically"? As an example of what I am talking about, the book "Structure and Interpretation of Classical Mechanics" (a cousin to SICP) tries to teach classical mechanics with the help of computer programs. But this approach remains unconventional.
    - davidivadavid 10y ago
      I should probably not have been sarcastic, especially when I agree completely about the importance of looking at math with a more constructive/computational point of view. Now, I'm not a mathematician or a scientist of any kind, merely a dilettante textbook reader, but I'm not sure that approach is sufficient, or even most efficient to teach all subjects. I think it arises naturally when you approach subjects where it's directly relevant: engineering, complexity theory, type theory, etc. and people who will need those insights will get them in due time. But yeah, broadening the audience could help people get a better appreciation of the connections between pure and applied math.
SomeStupidPoint 10y ago
You responded to a strawman of what I said, made no points that even remotely address or refute what I said, and further, decided that you were going to personally attack me. In fact, your link generally agreed with my point! (In summary: traditional reading and lecture methods have benefits; videos and other newer media can help (some) people, and give a new perspective; computers have a few useful behaviors; engaging imagination and exploration are better than rote learning.) That's really kind of pathetic, and a disservice to your view. I'd be happy to speak with you, though, if you want to actually read my comment and address a (charitable) interpretation of what I said, rather than your strawman one.

by dxbydt 10y ago

dunkelheit 10y ago

Traditional math education: a professor explains proofs of selected theorems on a blackboard and students are then required to memorize the proofs and reconstruct them during the examination. They are also required to solve a lot of problems that involve proving some consequences of the main theorems or doing some rote calculations using nothing but pencil and paper. The advantage of this approach is that it works. You can acquire solid intuition this way. But it is arduous as hell and also pretty one-sided - problems are limited to what an average undergraduate can do by hand. Now, when I think about using computers in math education I don't think of some "javascript doodad" with moving sliders. I imagine instead implementing an algorithm or doing a numerical experiment, learning about possible pitfalls in the implementation, running the algorithm on some real data, seeing it fail etc. I think this experience is pretty valuable too and can nicely complement traditional approach.

davidivadavid 10y ago
That's why that approach has been offered in courses traditionally called "Numerical analysis" that have been available to math students for... how long?
- dunkelheit 10y ago
  Taking the sarcasm out, your point is that such thing exists. Agreed. But usually it is just a single course, it does not cover everything. And why it is a separate course at all? Why should you study some field with pencil and paper and then (possibly, after a couple of years) study a subset of that field "numerically"? As an example of what I am talking about, the book "Structure and Interpretation of Classical Mechanics" (a cousin to SICP) tries to teach classical mechanics with the help of computer programs. But this approach remains unconventional.
  - davidivadavid 10y ago
    I should probably not have been sarcastic, especially when I agree completely about the importance of looking at math with a more constructive/computational point of view. Now, I'm not a mathematician or a scientist of any kind, merely a dilettante textbook reader, but I'm not sure that approach is sufficient, or even most efficient to teach all subjects. I think it arises naturally when you approach subjects where it's directly relevant: engineering, complexity theory, type theory, etc. and people who will need those insights will get them in due time. But yeah, broadening the audience could help people get a better appreciation of the connections between pure and applied math.

SomeStupidPoint 10y ago

You responded to a strawman of what I said, made no points that even remotely address or refute what I said, and further, decided that you were going to personally attack me. In fact, your link generally agreed with my point! (In summary: traditional reading and lecture methods have benefits; videos and other newer media can help (some) people, and give a new perspective; computers have a few useful behaviors; engaging imagination and exploration are better than rote learning.) That's really kind of pathetic, and a disservice to your view. I'd be happy to speak with you, though, if you want to actually read my comment and address a (charitable) interpretation of what I said, rather than your strawman one.