It is clear that Wolfram Alpha will help revolutionize the teaching of mathematics because of its impressive power and its ability to show the steps to solve mathematics problems. For example, you can get Wolfram Alpha to show the steps of solving the equation 2x^2+4x-5 = 12 by completing the square. I'm not sure how to get it to show the steps for solving this by the quadratic formula.
I may write more comments later about Wolfram Alpha but at least wanted to get this posted to try to begin a discussion about it.
I was reminded of Wolfram Alpha recently when I had used to it to compute the values of the prime counting function pi(x) and Gauss' logarithmic integral function Li(x) for selected large values of integers x. I am planning to give a talk soon to students at my school about the nature of mathematics as mathematicians see it, and I'm using Gauss' investigations about the prime counting function, his estimate Li(x) for pi(x), and his conjecture that Li(x) > pi(x) for all x (which is false) as an example of how inductive reasoning arises in mathematics. I had wanted to compile a table of values to help illustrate this example, and I remembered that Wolfram Alpha could help me (I was at home working on this, and I don't currently have a CAS at home).