Well, I must admit that times have changed, and it seems it is now the right time to be in tune with the wonders that science and technology have to offer. The advent of immensely powerful Computer Algebra Systems (CAS) is giving us more and more reasons to switch out teaching styles to a new paradigm, where the ideas are presented together with real visual computer-generated representations, and where the emphasis is put on the concepts rather than the symbolic manipulation.
For example, in a typical first year college Calculus class, it takes a great deal of effort to go over a series of techniques that help the students to understand integration. Those techniques are clearly mechanical and repetitive, but yet students have a hard time understanding the main ideas. Nowadays, software like Mathematica and others are capable to solve symbolically some very complicated integrals, which go way beyond what an accomplished first year calculus student can do.
Shouldn’t we make an emphasis on the concepts rather than on the calculations? In mathematics, it is hard to separate because the two go tightly together. But I certainly believe that we would benefit by introducing systematically the use of CAS in the classroom. There’s a trend in most of the colleges to introduce computer assignments, as a part of the curricula, but from my experience, students are not getting most of it. They still don’t see the computer as a friendly ally at the time of learning math. But yet, they would gladly settle for a calculator.
The future of CAS should also include a way to use all this "intelligence" used to solve complicated problems to also being able to "explain" how to arrive to the answer.
