This is a point of view that I'm hearing a lot now, mostly from technically capable people who know that computer algebra systems exist and are more reliable than doing everything by hand. And there is merit in the argument, but I've always felt uncomfortable about it, as if something was missing.

More recently I think I've identified what it is, and I included a little rant about it in my blog post about the birthday problem[0].

In particular, you've said:

    > I wonder if there's not some way
    > to skip a lot of the tedium of
    > algebraic manipulation that is
    > forced upon students,

I'd like to compare this with the idea of missing all the tedium of practising the cross-court forehand drive in table tennis. And the answer in that case is no, not if you want to be a top flight player. You need your body to recognise the shot automatically and play it without thinking, so your brain is released to do the higher-order stuff necessary to work on the problem, not the detail.

But more than that, sometimes it's the hours of practice in algebra (or similar) that means that when something turns up in disguise then you still recognise it, and still know how to torture the equations to twist them into the standard form.

It's really hard to explain. Sometime I'll have another go at it, try to put into words the meta-intuition I've developed over the past 40 years. In the meantime, the side-box with the rant is the best I've managed.

[0] http://www.solipsys.co.uk/new/TheBirthdayParadox.html#toc_na...