> 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 don't think, they calculate. Use computers for calculations. To understand mathematics, don't use a computer; use a text book. If you don't understand mathematics, then you can't be thinking but only guessing. That's what I've seen students do, to use a calculator as a guessing machine, checking their guesses until they get an answer they guess is correct.