The issue for me is whether students can manipulate algebraic expressions with purpose, depending on context, understand that they can be pulled apart and put together, and so on. The students I get now in college know (if they and I are lucky) a few standard techniques that they *always* use whether or not it's appropriate (e.g. they never write 1/sqrt 2, because "you never put a square root in a denominator" -- somehow they're convinced that (sqrt 2)/2 is correct, and 2/sqrt 2 isn't), but generally do not have what we might call algebra sense -- the same understanding of rational functions that they have of rational numbers. This means things like the general notion of factoring and some facility outside of basic quadratics, simplifying expressions when desirable, complicating expressions when desirable, common denominators and common multiples, primes, and so on.
I would guess that this algebra sense can only come about by their own manipulations of algebraic expressions, but don't see the calculator as the enemy -- as always, wise use of technology may be helpful. If anything, technology can allow us to shift away from hammering away on the 2 or 3 techniques we've always hammered away on and lift the vision a bit.
Of course I don't teach at this level myself, so maybe I'm just full of hot air. I'd be interested in hearing from the folks who do teach this, from upper grades in elementary on -- how do you develop this "algebra sense"?
==================================== Judy Roitman, Mathematics Department Univ. of Kansas, Lawrence, KS 66049 firstname.lastname@example.org =====================================