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I may be old school, but I believe that technology can only very mildly help with mathematical understanding. Math is build of bricks, one on top of the others.

I may be old school, but I believe that technology can only very mildly help with mathematical understanding. Math is build of bricks, one on top of the others. And to understand advanced concepts, you need a deep and solid understanding of the lower layers. I think you only get this understanding by thinking hard, looking at examples and building your own. Calculators are fine if you are dealing with numbers, but if you work in a quantitative field you often don’t deal with numbers directly: you deal with expressions involving variables. You manipulate the expressions, and at the very end you plug in numbers. A calculator is of no help, and I hope you did not use one when you learned how to deal with numbers. You better be able to manipulate fractions, and know the distributive & associative laws, and know when to complete the square, and the exp(ln(x)) trick, etc… What, you say I could use Mathematica? Of course, and I do -- when I know exactly what to compute. But generally I do not: I have these relations and I try to make sense of them. Moreover, when the final result is nice, concise and elegant, it means that I do not fully understand the problem. Examining the computation will help me understand what is happening: what part of the equations cancel with what other part, and do I understand why? Good luck doing that with Mathematica. Another example: continuity. Nothing is simpler: you plot a few graphs, and the functions are discontinuous where there are jumps. Who need this epsilon/delta gibberish? - Functions that are discontinuous everywhere? Ok, I can wrap my head around that. Still no need for epsilon deltas. - A function that is discontinuous on the rationales and continuous on the irrationals [1]? Good luck understanding that with your graphic calculator. - And the topological definition of continuity [2]? This is a beautiful definitions, that unifies all the definitions of continuity you have seen for all these functions of THIS space into THAT space. Well, thinking and well chosen examples are going to help you understand the definition, not technology. [1] http://en.wikipedia.org/wiki/Thomae%27s_function http://en.wikipedia.org/wiki/Thomae%27s_function [2] http://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces http://en.wikipedia.org/wiki/Continuous_function#Continuous_...

Dn_Ab 15y ago
I must sincerely and strongly disagree with you. Technology can help elucidate mathematics beyond plug and chug. Teaching is one of the best ways of learning. a rough paraphrase of something I read someone say is they write a new book every time they wish to learn something new. Writing a computer program which embodies a mathematical concept is teaching the most retarded entity that is capable of more than just arithmetic. Certainly anything which is non-constructive falls outside of this, but in laying a motivation and providing a foundation that reduces the amount of problems you need to do by say an order of magnitude? Technology is unmatched. Although the requirement on constructive* maths seems restricted you would be surprised that both your [1] and [2] level of abstraction can be tackled with such tools. http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf, http://haskellformaths.blogspot.com/ http://haskellformaths.blogspot.com/, http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell.html http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell... You do it this way, vary enough examples and try to anticipate results, you will develop a number sense that is required to be comfortable with maths. It worked for me. * I am skeptical in the reality of arbitrarily real numbers because I am skeptical in the reality of hypercomputation.
- slowpoke 15y ago
  Writing a computer program which embodies a mathematical concept is teaching the most retarded entity that is capable of more than just arithmetic. I have to agree with this. Solving a problem - this isn't even limited to just math - with a computer program often means to generalize it. Generalization requires understanding. Thus, if you manage to generalize something, you have understood it.
thaumaturgy 15y ago
I haven't been able to make up my mind about the role of technology in education (especially computers). On the one hand, given that he's a passionate scientist and educator, I take Clifford Stoll's opinion seriously when he says that he believes computers don't belong in classrooms. I think that the effects on developing brains of large amounts of time with computers is undeniable at this point. And, as an observer, I do not see that the greater and greater application of computers in classrooms is resulting in more intelligent students. (More informed, possibly.) On the other hand, a very long time ago, I had a cousin who was visiting and was having trouble with some physics homework about the Doppler Effect. Being a proper nerd, I quickly whipped up a simple animation of an aircraft at varying speeds across the screen, with concentric circles at regular intervals. I thought I understood the Doppler Effect and sonic booms and all that; when I was done, and I saw it in action, I realized how poorly I'd actually understood what was going on. So ... in the lower grades at least, I think there's a chance that computers could help with mathematical understanding. Younger students seem to have a lot of trouble with things like mathematical association and commutation, which seems to lead to a longtime discomfort with things like the mental math described in the article. Maybe some kind of multimedia demonstration early on would help that.

Dn_Ab 15y ago

I must sincerely and strongly disagree with you. Technology can help elucidate mathematics beyond plug and chug. Teaching is one of the best ways of learning. a rough paraphrase of something I read someone say is they write a new book every time they wish to learn something new. Writing a computer program which embodies a mathematical concept is teaching the most retarded entity that is capable of more than just arithmetic. Certainly anything which is non-constructive falls outside of this, but in laying a motivation and providing a foundation that reduces the amount of problems you need to do by say an order of magnitude? Technology is unmatched. Although the requirement on constructive* maths seems restricted you would be surprised that both your [1] and [2] level of abstraction can be tackled with such tools. http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf, http://haskellformaths.blogspot.com/ http://haskellformaths.blogspot.com/, http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell.html http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell... You do it this way, vary enough examples and try to anticipate results, you will develop a number sense that is required to be comfortable with maths. It worked for me. * I am skeptical in the reality of arbitrarily real numbers because I am skeptical in the reality of hypercomputation.

slowpoke 15y ago
Writing a computer program which embodies a mathematical concept is teaching the most retarded entity that is capable of more than just arithmetic. I have to agree with this. Solving a problem - this isn't even limited to just math - with a computer program often means to generalize it. Generalization requires understanding. Thus, if you manage to generalize something, you have understood it.

thaumaturgy 15y ago

I haven't been able to make up my mind about the role of technology in education (especially computers). On the one hand, given that he's a passionate scientist and educator, I take Clifford Stoll's opinion seriously when he says that he believes computers don't belong in classrooms. I think that the effects on developing brains of large amounts of time with computers is undeniable at this point. And, as an observer, I do not see that the greater and greater application of computers in classrooms is resulting in more intelligent students. (More informed, possibly.) On the other hand, a very long time ago, I had a cousin who was visiting and was having trouble with some physics homework about the Doppler Effect. Being a proper nerd, I quickly whipped up a simple animation of an aircraft at varying speeds across the screen, with concentric circles at regular intervals. I thought I understood the Doppler Effect and sonic booms and all that; when I was done, and I saw it in action, I realized how poorly I'd actually understood what was going on. So ... in the lower grades at least, I think there's a chance that computers could help with mathematical understanding. Younger students seem to have a lot of trouble with things like mathematical association and commutation, which seems to lead to a longtime discomfort with things like the mental math described in the article. Maybe some kind of multimedia demonstration early on would help that.