What are the advantages and disadvantages of the Moore method? Describe your experiences with the Moore method. What are its advantages and disadvantages?
 A: I think the Moore method is a very inefficient way of imparting information. On the other hand, students profit by having a class taught by the Moore method, since it forces them to think in a different way. My view would be that Math majors should take one (but no more than one) class taught by the Moore method.
A: I taught a course on Galois theory at Canada/USA Mathcamp.  The context was certainly different from a regular university or college: there were no exams or evaluations of any kind, and students choose their own courses and drop casually out of them if they wanted.  I started with 28 students (many of whom did not belong there), and ended with only 14.  I think that 10 out of those 14 had learned a significant amount of material rather than being over their heads, and 4 out of those 10 understood everything we had done perfectly.  When you compare this with the outcome of teaching a course on Galois theory with traditional method, it is not bad at all.
It is true that the material is covered slower than we traditional lecture, but I believe that students learn more.  The students who finished the course were certainly enthusiastic and I had the same experience that Kevin tells in his last paragraph.
Finally, I will add that a competitive atmosphere is not necessary for the Moore method to work. (Yes, Moore's original course required aggressive and competitive students, but there are many things we call Moore method now.)  In my case, it was more of a community working together.  I recall how, after proving the fundamental theorem of Galois theory, a student was attempting to compute $\cos \frac{2\pi}{17}$ explicitly (in an afternoon, preparing for the class for the next day), and he was surrounded by a group of other students, eager following the process and cheering him on.
In short, after my limited experience, I am a big supported of the Moore Method and other variations.  Particularly for students who want to go on to become mathematicians, it gives them a more realistic taste of what math is than a traditional course.
A: The Moore Method has other variants grouped together under the "IBL" (Inquiry Based Learning) umbrella. I just finished teaching an intro (undergrad) real analysis course using an IBL script instead of a textbook and am amazed at how much more solidly all of my students absorbed the information in the course. 
Certainly, this method doesn't move as quickly as traditional methods, but students assimilate the information so much more fully this really doesn't matter. 
BTW: If you are interested in resources for running such a course, there are refereed IBL scripts in the journal of inquiry based learning in mathematics. 
A: I am a very latecomer to the question. Thus, I doubt my contribution will do any good. I try. The Moore Method is one of my favorite "teaching" methods. I have used a variant of it several times and in particular in two courses: Multivariable Calculus and Number Theory. For the former, I designed the course myself (for details see my paper "Moore and Less" : http://www.tandfonline.com/eprint/jGE3QNxcuGzUGj273smp/full), For the latter, I used "Number Theory Through Inquiry" (http://www.maa.org/ebooks/textbooks/NTI.html). Generally speaking, the advantages have been mentioned more or less in the previous answers/comments. Thus, I focus on potential disadvantages. 
Playfulness: The Moore Method strictly used is not that much playful. The point is you have a setting in which the materials have been prearranged. This forces you and your students to stay on a predesigned track, and as a result, hinders useful and playful jumps. Let me give an example.  Suppose you start with some examples of primitive Pythagorean triples. The most famous ones are (3, 4, 5) and (5, 12, 13).  Observation: the difference between two of the numbers is one.   Having characterized Pythagorean triples, it would be natural to move to Pell equation.  However, since the materials have been prearranged, you should continue with Lemmas ., Theorems  . , all directly related to Pythagorean triples. 
Naturalness:  Again, this is somehow related to the prearrangement.  As a designer, you feel that you should provide some backgrounds to help students to prove a certain theorem. What you provide is natural for you since you already know the proof of the theorem.  However, it is not so natural for most of the learners. Let me go with the previous example.  Moving towards the theorem characterizing Pythagorean triples you write (your students read): “It turns out that there is a method for generating infinitely many Pythagorean triples in an easy way. It comes from looking at some simple algebra from high school. Remember that … “.   It gives me a very bad feeling to behave with my students in this way, to say the least.  The problem arises even for the lecturer when others have designed the course.  An arrangement that is natural for somebody else is not necessarily natural for you. 
Forcefulness:   Your students are forced into forward thinking.  There is a theorem (again take the previous one as an example). You are somehow sure that your students need some help to prove it. Where do you provide that help?  As a Lemma before the theorem! 
Connectedness: This one is very strange and paradoxical. While you are connected with individuals and/or small group of students working together, you lose your connection with the class as a whole.  
There are some other points. I stop here since my answer is already too long. Moreover, I couldn’t find suitable words ended with “ness”  to describe  the other points ☺ 
A: One thing that should be mentioned is that the "Moore Method" is not precisely defined. There are many variations and many selections of students and professors.
Along these lines, I've taught a "self-paced" calculus I course twice. The students were given a massive load of homework problems, and each student had their own homework (everyone had "differentiate these polynomials", but each student had their own polynomials). I encouraged working together, and we spent most of the lecture time working homework in small groups. The exams were generated automatically, and the students had access to many "sample exams". Each student took the exams during office hours, at a time of their own scheduling (with a deadline). The homework was graded through Webwork, so that students could easily email me questions about particular problems. I received about 1200 emails the first semester, and around 800 the second.
The student evaluations for those two sections are the worst student evaluations I've ever received. In part, the stock questions are more geared to evaluating a traditional section, but mostly I think this was because the course was a lot of work for them. Both semesters, about half the class used all that extra rope to hang themselves (metaphorically). One semester, there was a group of 4 students who became competitive, in a friendly way, and became 4 of the best students I've seen at my current institution.
But perhaps the most telling factoid is this. In five years at this school (teaching perhaps 20 courses), I've had three students go out of their way to find me later and say that my course was the best course they've had, the one they learned the most in, the one that set the tempo for the rest of their studies. Not asking for a letter, mind, just wanting to say "thank you". Only one of the three was a math major. Two of the three students were from the first time I taught that way, and the third was from the second time.
A: I took a class using the Moore method during my freshman year as an undergraduate (back in 1998).  I really enjoyed it, but now that I've acquired a bit more teaching experience I think the following points should be made.  I should remark that I've never tried teaching a course using this method myself.
1) Like Felipe said, it is not good for conveying a lot of information.  It is more of an ``experience''.  It is thus best for classes like baby-Rudin style analysis classes in which there are few hard theorems and not too much that HAS to be covered.
2) It is important that the class is small and that the experience/talent levels of the students are relatively equal.
3) I think that such classes can be valuable to a certain kind of student (one who is relatively strong already and who enjoys competition), but I don't think it should be mandatory.  Maybe one section out of a multi-section class.
I want to elaborate a bit on the third point.  I think the stereotype of mathematicians as aggressive and hyper-competitive is a dangerous one, and has the effect of discouraging students who do not fit that stereotype.  It is important that when a course like this is offered, an effort is made to clarify that not all students need to take it.  I could envision a culture developing where the Moore method section has a reputation as being the "real section" for the "best students".  The faculty need to make a serious effort to prevent this from happening.
