Hi,
I was enjoying your blog postings about Math. I particularly like and agree with this post http://sites.math.rutgers.edu/~zeilberg/Opinion128.html
I think your point about giving a concrete, non-trivial example of the details in solving / proving a step is absolutely the biggest change that math teaching could ever make!
I remember being very frustrated with my math classes... It wasn't until my third semester of calculus that I felt like a had a small grasp of the material from the first semester. It wasn't until I did a lot of work in visual effects that I gained a good handle on calculus, which is a bit sad really.
I was in college 20 years ago. I just recently took a course in cryptography online. It was great except I found it so very frustrating that the lectures focused on symbol manipulation and were very lacking in concrete explanations of the steps in solving a problem. At work when I design software I'm explaining to my coworkers "why" and "how" I'm putting the pieces together. I'm helping them see the sequence of steps and why I'm doing things the way I am, and almost always with examples so they can see how the execution steps flow and how state is built up and changed. In contrast to this, during lectures several times the phrase "I hope everyone sees that the answer has to be X" would be used and then maybe a short expression would be written done, but without ever building up the underlying sequence of steps.
I can understand the argument that professors want to avoid training students to just follow a sequence of mechanical steps without actually thinking and reasoning through what they are doing, but currently it's the other extreme.
Along with giving a concrete example and how to solve a problem, the other time I learned the most was when a professor would explain some of the underlying principals for why a step was being followed or what an operation being usable implied about the problem at a fundamental level.
Hmm, after writing all this, I wish I had some math problems along with their solutions to help illustrate what I'm talking about. I'm afraid I'm being a bit abstract here.
In any case. I'm enjoying your blog.
Cheers, -Brian Leair