On 2013-11-12, Hetware <firstname.lastname@example.org> wrote: > I'm working through a 1953 edition of Thomas's _Calculus And Analytic > Geometry_. When I work problems, I use Mathematica to type my > transformations, and to check my results. I use it for far more, as > well; graphing, numerical solutions, etc.
> Many years ago I found computers to be a nuisance when it came to math, > and more importantly physics. I was contented to have a piece of chalk > or a pencil and an eraser, than to have all the computing power in (the) > Universe. Time was the only resource I found in short supply.
> Now that I have used them for years, I realize that computers can do a > whole lot. They can find integrals for equations which I cannot > integrate by hand. They can produce graphics which a human could never > produce, etc.
> I've used a pocket calculator since the 1970's. But, I feel as if I > should have learned to work the same problems on my own. I feel > somewhat crippled by using it as a crutch.
> I'm in a conundrum twixt the use of computers to do my thinking for me, > and learning to think for myself. Should a child learn his times > tables, or learn to use a computer to do it for him?
Computers can do things which they have beeen programmed to do and nothing more. I have found that it can be somewhat unwise to trust computer programs too much. I have found errors in them, most recently yesterday. Also, mathermaticians keep finding integrals which the current computer programs have not been able to find. One can produce a computer progrram to find the integral of any function integrable in terms of elementary functions, but this can take a looooong time, so short cuts have been devised, but they can give up.
As for your last paragraph, there are advantages to being able to do some things yourself. It takes time to use the computer. Also, there are things you can find out which you would not expect doing it yourself. Getting the result yoorself often leads to more understanding, but not always.
As for what should be learned, memorization is useually a poor way to learn anything, and might leave out alternatives. I would expect a child to learn how to construct addition from counting, and multiplication from addition, and to produce the tables for himself or herself. This would also provide a quick means of filling in a forgotten value.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University email@example.com Phone: (765)494-6054 FAX: (765)494-0558