Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, i.e. professors lecture to students and sketch some proofs. Suffice it to say I'm not a fan of the current methods and I don't think it would be too far from the truth to say that I do all the actual learning outside the classroom. Has anyone tried anything different and seen any difference in student understanding and comprehension in graduate or undergraduate courses?
Some background motivation: I'm a TA and my current method of doing things is to just write some problems on the board and then go through their solutions. This is fine and it's what the students expect but sometimes I feel guilty because I'm just teaching them problem/solution patterns and reinforcing all the bad stereotypes about what mathematics is instead of showing them the underlying conceptual tapestry and helping them rethink their attitudes toward mathematics. It's kinda like the old saying “Give a man a fish; you have fed him for today. Teach a man to fish; and you have fed him for a lifetime”. So basically I throw a bunch of fish at the students hoping it will feed them for the semester.
Best Answer
The topic you touch upon is vast, but I wanted to comment on this phrase: "problem/solution patterns which is very different from showing them the underlying conceptual tapestry".
If for some reason you have to use this format (department restrictions or whatnot) choosing your problems well will simultaneously introduce some of the conceptual tapestry. Rather than introducing a mathematical tool and then the problem that goes with it, you introduce the problem first (just out of range of the student ability) and bring it to the point where things get stuck, where something new is needed to go further. Then the motivation is clear for the new tool.