"Hetware" wrote in message news:d6ydnaKnDM2PNBzPnZ2dnUVZ_tqdnZ2d@megapath.net...
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? ======================================================== Can you pilot a plane? Can you strip down a jet engine and reassemble it? Can you calculate the optimum shape of a wing?
There is no rule that says you have to do all three to be competent at any one of them, nobody can do everything. We all rely on the competence of others and have some specialty ourselves. I'll trust a computer to fly a plane before I'll trust myself or a human pilot, just as I'll trust a calculator not to make an arithmetic mistake. The child should learn algebra instead of tables, the real problem is the teachers are not competent to teach algebra so they bore the kids to death by teaching tables by rote and the kid grows up hating math, which is a handicap. If the teacher had learnt algebra at age 7 they'd be teaching kids algebra at age 22. You have to teach to principle of multiplication, count = row * column, but not tables. Arrange 7 rows of 9 columns and count, 63. 8^2 = 64, (x-1)(x+1) = 7 * 9 = 63 = x^2 -1 4^2 = 16, (4-1)(4+1) = 4^2-1 = 15 That can be learnt by rote instead.