The points Stephen Wolfram is raising are very relevant and I like the idea of using the computers more and in a different fashion for teaching mathematics. There is one other issue with teaching mathematics (and most sciences) which I think is also very important: the students are first thought the methods for solving the problems and only after they are presented with the problems to solve. This practice always seemed very unnatural to me. I think they should be first challenged by the problem and then helped to "discover" the methods to find the solutions. I know this is not always easy, but we should at least make an effort. It would make learning more of a challenge, than a dull task. The most frequent question/complaint I heard from students both as a student and later as a (part time) teacher was “why do we need all this?” Whenever I was able to give a clear answer to this question it worked like magic in creating interest and involvement in the class.
Bela Farbas

I remember asking once what tangents were used for when we were studying trigonometry at high school. And there wasn't an answer. Not on the curriculum apparently to know what it is used for.
But studying Engineering at University was different. There were a lot of real world applications. Calculus didn't make sense until University. The math was more advanced but the examples were better or more relevant and it made sense. It wasn't until 3rd year tensor calculus that I was again doing math that wasn't attached to a practical outcome.
However I don't think it is just an issue of using a computer and then it becomes easy. And we do want to ensure people understand the problem and the mathematics options that can be used to solve it rather than how to use a computer program and punch in the numbers. I think the computer option reduces maths back to the realm of magic where you know the right incantation and get the right outcome.
Not all learning modalities will deal with this the same anyway. We need more than visual representations.
I do agree that making the subject engaging and relevant is the key to students embracing it.
Ray Keefe

I like the comment about incantations, Ray. I think that if the problems are set in the right way, then the student will need to understand what needs to be done (and why) so that they can then choose the right approach. If that isn't done, then I think your concern is a very real one.

In my studies, I have personally found that computer interactions are much more effective in teaching me the concepts of the subjects rather than calculating by hand.
Ray, you mention that you think using computers will reduce maths back to magic, but the true underlying reasons for maths is generally not understood by students anyway. All we are trying to do is get the answers right and be able to answer the exam problems. Like it or not, thats how it is.
One example is when I was studying control engineering. I understood as much about how the rlocus command in matlab worked as I did the 12 steps to hand drawing a root locus. The only difference, I could experiment easily with the matlab command and because of this I was able to learn what happens to a system when I add poles or zeroes quickly. This understanding helps incredibly when trying to make a lead-lag compensator.
Now, the question we need to ask if what would I have gained if I did this by hand. The simple answer is a cramped hand and less sleep.

On another point, problem solving is (generally) implemented horribly in any maths based subject I have taken. There are generally two tactics taken. One is to set pages and pages of unrealistic, oversimplified problems in which we will just wrote learn how to solve each and every one, to the point where we don't even need to read the entire problem in an exam in order to know how we are "supposed" to be solving it. Second is to set problems that require deep thought to get an answer. This is fantastic when it is matched to the students level of knowledge, but it rarely is. It is easy to forget how hard things have been when you did them a long time ago. When I reflect on my first year classes, they seem infantile and I can't believe I didn't ace every single one, but at the time they were tough Lecturers often have the same affliction. "I can solve this, why can't you". A nice middle ground is needed but it is hard to find, but through in depth student feedback it should be achievable.

Computers simply let me experiment without the worry of hours of lost time, and cutting down rainforrests of workbooks full of equations.

But everyone works differently, your millage may vary.

I agree that computers can be a wonderful tool for exploring the mathematical responses of systems to parametric changes. I use them this way all the time. So this is a big plus and should definitely be used to the max.
But the simulation doesn't always help you understand the implications in the real world. This is harder.
As an example, I became an expert in acoustics and the design of high performance amplifiers through listening to the results of hours of experiments in the design of high performance audio preamplifier and amplifiers. No amount of theory on THD and phase distortion could have given me the same insight.
Engineers solve problems in the practical domain.