Remember how calculators came and took the bite out of slide rulers. In a world far, far away, e.g. Eli's bunnyhood, engineering students took classes in how to use the things and adding machines were treasures. In small groceries people used brown paper bags, in large ones, very clunky cash registers.
At that point, still lamented in teachers and faculty lounges world over (and, as Brian would point out, in the US Republican Party), actually knowing how to do arithmetic became less of a necessity and more of a party trick, what was needed was the ability to set up the problem and punch the numbers into the $1 calculator, $10 if you have to do logs or trig functions.
Thoreau, who Eli spars with in a friendly manner, brings news of the next frontier. Students everywhere rejoice, calculus class is no longer needed.
I usually use Mathematica as my calculator, especially when explaining homework in office hours. I can refer back to variables and quickly make graphs or manipulate symbols. So while going over homework with students in my biophysics class, I pull up Mathematica and one of them says “Is that just like Matlab?” My eyes bugged out. A biology major who knows Matlab? This is the subject of my interdisciplinary hopes and dreams. So I asked what she uses Matlab for. “My boyfriend is a mechanical engineer and I have to do his homework for him.” My reply was “Your boyfriend should do his own homework, and if you’re a biology major with Matlab skills you should be working in my research group.” . . . . . . . . (Yes Chip, Eli left out a paragraph, go to the link and see if it changed the meaning of the piece)Sort of like most students can't remember long division, why should they, all they need is to be taught MatLab (the affordable version) or Maple or Mathematica (the high cost ones) and to find a function. The same thing happened with statistics, all you need is R or SAS or Statistica. To the joy of mathematicians and statisticians they now only have to teach the lemmas.
Contrast this with my discovery yesterday in biophysics: I gave them an activity in which one step required that they calculate the derivative of a*x^2-b*x. I know that calculus isn’t a prerequisite for the course, but they’re all juniors and seniors and the biology department requires them to take a quarter of calculus. Alas, most of them could not remember it.
Symbolic algebra/calculus programs are the calculus versions of calculators. A light has dawned. Death to ten blackboard long proofs.