Part of the problem with this is that software like Mathematica or even SAGE is quite opaque. So you can teach folks what a derivative is, tell them that a black box called Mathematica/SAGE will calculate derivatives for them (as long as they don't hit a bug in the implementation) but the connection between the two is lost, unlike with teaching the calculation algorithm as part of the course.
AFAICT, the best way of preserving this connection while mechanizing the calculational parts of math is to employ proof assistants such as Coq and Matita, which can extract a mathematical algorithm from a formally-verified construction, and run it on input data. However, both Coq and Matita are currently limited to rudimentary tasks, such as manipulating expressions in simple algebraic structures such as rings and fields. They are a long way from having even the equivalent of a simple CAS, partly because CAS themselves have little in the way of formal foundations--this is quite evident as soon as one tries to deal with such issues as branch cuts, real vs. complex quantities, provisos, dummy variables and the like.
In short, this idea has plenty of long-run potential but is not quite ready for prime time.
Is what you want more like the output of Wolfram Alpha? If you ask it to compute a derivative, integral, or limit, it shows you the steps it takes to get there.
TL;DR= There doesn't exist a course/curriculum/general textbook based on Conrad Wolfram's "Computer-Based Math Education" idea. Let's create an open-content one! .... if we can
By computer-based math, I don't mean "math as usual, now taught through a computer!" (a la Khan Academy) I mean "math where we let computers do the calculation drudge-work, while we do the interesting parts."
Or, paraphrasing Conrad Wolfram: "stop teaching kids how to take derivatives; that's what MathematicaTM is for. Just teach them what a derivative is, so we can move on to more interesting problems. Like, you know, the ones in the real world." (Here's Wolfram's original polemic about the issue.)
Obviously, this is controversial, and Wolfram spends most of his talk rebutting arguments against it. If, after reading them, you're still not convinced that this is a good idea, then start another thread to discuss it. I don't intend this thread to become a blues-vs-greens battleground. Seriously, just start another thread.
On the other hand, if you are convinced that Wolfram is on to something...
My problem with this whole venture is that it's too important (IMO) to be left to the Wolframs.
I mean, come on. Wolfram's basic thesis might be true, but it's no coincidence that this particular truth is being spouted by the brother of the guy who created Mathematica.
And, unfortunately, the Wolframs seem to be the only ones pushing for it. Which means that we won't get any "math, not computation!" courses/textbooks until they can find a taker.
Now I'm guessing that most LWers would want to reap the benefits of Wolfram's basic idea without having to pay his family a fortune for it, and before however long it takes them to convince an education board about it. (How many "How do I go about learning useful math?" threads have we had so far?)
So why don't we give the world a leg-up on the path to the widespread mathematical literacy that Wolfram promises? Why don't we put out a computer-based math course for the world?
Obviously, we'd have to use free stuff... Sage instead of Mathematica, for instance. And whatever we put out would have to be free, because... well, if you could write textbooks that people are likely to pay for, you wouldn't need to be part of an LW community venture to do it.
My major questions, therefore, are:
Are there enough (a) mathematically literate LWers with (b) tons of free time who (c) think computer-based math education is a good cause and (d) are willing to work for free toward a good cause?