effective teaching 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.
 A: I've seen Mazur speak on this and he says a fair number of his students don't like it.  Some of their reasons for disliking it apply more to most other institutions than to Harvard.  If they've always been taught that learning mathematics consists of memorizing algorithms to apply to assigned problems, they may regard a more intelligent approach as grounds for complaint, especially if the reason they're there is to get an "A+" in the course so that they can forget about it and get admitted to medical school (admission to which requires calculus or the like, not because students need to know that subject, but because they need to have demonstrated that the can succeed in courses that are challenging).
So there's some danger of being punished for doing the right thing.  That doesn't mean you shouldn't do it, but it might affect the way in which you do it.
A: You may also wish to look into the book How to Teach Mathematics by Steven G. Krantz. Perhaps Section 3.10 on the teaching reform and the references cited there could provide a reasonably good starting point for addressing your question.
A: I just watched the video -- I didn't know of it and it is certainly very interesting and provides much food for thought.
From the perspective of someone who teaches, I could relate to parts of it: particularly the decreasing ability to understand student's questions.  From the perspective of someone who was a student, though, I do remember learning during lectures: perhaps not in all courses, but certainly in some which I still remember fondly.  These courses were usually lecture-based; although in one case the lecturer would ask questions to the students all the time: picking a starting person and then moving systematically along the audience.  This used to instill the "fear of God"  in some people, but it meant one had to be on top of the material.  I enjoyed that and, in fact, it boosted my confidence.
Now to answer the question, in the University of Edinburgh (where I am based) we started a few years ago to teach some of the introductory courses incorporating some element of Peer Instruction.  I personally have not taught introductory courses for a while, so I cannot say how this is panning out.  The School of Physics (I'm in Maths) has been teaching the first-year introductory course using Peer Instruction for some time now and they seem to be very happy with the result.
I wonder whether some variant of this method can work for final-year courses, though.
A: I've been to a couple of courses where the lecturers asked students to prove many fundamental theorems and examples as exercises (eg Ravi Vakil's algebraic geometry, you can see this in his online notes http://math.stanford.edu/~vakil/0708-216/ ). I know some students find it very frustrating, and for me this style is time-consuming and -inflexible (as in, if I don't try the questions straight after the lecture, I can't understand the next one), but I came away with a much deeper understanding than I usually get from courses. (I haven't heard of the Moore method before, but this sounds like something related.)
A: This very stimulating presentation on teaching math effectively came out in May 2010 and addresses some of the issues presented in the other answers:
Dan Meyer: Math class needs a makeover
Although applied to high school math, at least some aspects of the technique could be incorporated, if only in a few sessions, into advanced classes. Comments to the video by educators and students provide some feedback on the technique.
Another potential method for revamping math classes for the 21'st century (maybe start viewing at time stamp 6:50):
Salman Khan: Let's use video to reinvent education
And always a good read and reminder: V.I. Arnold, On teaching mathematics
A: 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.
A: You might like to see the argument in 
What should be the context of an adequate specialist undergraduate education in mathematics?, Ronnie Brown and Tim Porter, The De Morgan Journal 2 no. 1, (2012) 411--67. 
http://education.lms.ac.uk/2012/04/r-brown-and-t-porter-what-should-be-the-context-of-an-adequate-specialist-undergraduate-education-in-mathematics/
and the comments that followed.
Here we argue against the concentration on content, without any discussion about mathematics. For more discussion on such teaching questions, see also my web page http://pages.bangor.ac.uk/~mas010/publar.html
Some of our students rated our course on "Mathematics in context" as the best of our courses. Students had to write a project under the terms of that title; we were totally surprised by the initiative of the students and the variety of topics chosen; some also said that the writing helped them to come to terms with their own attitude towards mathematics. 
A: Mazur is a fine speaker, but he fails to mention an important thing which importance is underestimated - teaching good students. Students who are smart, interested and willing to learn can be just as easily harmed by bad teaching (or helped by proper teaching, for that matter). Sadly, many people seem to think that teaching good students doesn't require much attention or conceptual thought, since they would learn the subject by themselves anyway. Quite the contrary, excellent students require more attention precisely because they are more capable, so there is more potential to be tapped by a good teacher.
Teaching gifted students is, IMHO, more difficult than teaching mediocre ones, but also more challenging, since it's difficult to make general statements and it's not obvious what the right approach is (in the case described by Mazur, we more or less know the "right" way of doing things - understanding instead of memorizing, thinking vs mindless application of recipes etc.).
A: Have you heard about this before: https://en.wikipedia.org/wiki/Moore_method ?
I think it's pretty different to the ordinary teaching method, and might be one possibility that differs significantly from the standard teaching method. Basically, it seems like the students are given the basic theorems, axioms and such, and then asked to prove these theorems for themselves given the axioms, and construct examples for these axioms to familiarize themselves with the material. One thing I don't understand with this however, is how one manages to get through the course and cover a sufficient amount of content with this method (given that it naturally seems to take up a longer amount of time).
I think this could also apply to the way in which you might self-learn things from a book - perhaps instead of just reading the book, you could try proving the shorter theorems by yourself (instead of just reading their proofs), take a quick glance at the proofs of the longer theorems and fill in the details for yourself.
A: I found this on a search for the "Moore Method", used for point-set topology at the University of Washington in the late 70's.  Does anyone know where to find the (then mimeographed) hand-outs?  As I recall, there were less than 20 pages of definitions, axioms, lemmas, and theorems.  A beautiful thing.  There is much on the Moore Method floating around, and yet this concrete demonstration is nowhere to be found.  Help!
[2018 update]
The prof was Martin Bendersky.  I found him at CUNY and he remembers the class but can't help with the notes.
And yet, after all this time, I think I found them, following all the front matter here: 
http://zeta.math.utsa.edu/~zjo970/teaching/2012tfall/12f_top/top_bing_notes_student_V110323S.pdf
