Halmos, P.R. "How to Teach," I Want to be a Mathematician. Springer-Verlag:254-264 (quote from p. 258) (1985).

One of the rules was that you mustn't let anything wrong get past you -- if the one who is presenting a proof makes a mistake, it's your duty (and pleasant privilege?) to call attention to it, to supply a correction if you can, or, at the very least, to demand one.

The procedure quickly led to an ordering of the students by quality. Once that was established, Moore would call on the weakest student first. That had two effects: It stopped the course from turning into an uninterrupted series of lectures by the best student, and it made for a fierce competitive attitude in the class -- nobody wanted to stay at the bottom. Moore encouraged competition. Do not read, do not collaborate -- think, work by yourself, beat the other guy. Often a student who hadn't yet found the proof of Theorem 11 would leave the room while someone else was presenting the proof of it -- each student wanted to be able to give Moore his private solution, found without any help. Once, the story goes, a student was passing an empty classroom, and, through the open door, happened to catch sight of a figure drawn on a blackboard. The figure gave him the idea for a proof that had eluded him till then. Instead of being happy, the student became upset and angry, and disqualified himself from presenting the proof. That would have been cheating -- he had outside help!

Since, as I have already said, I have not really seen Moore in action in a serious mathematical class, I cannot guarantee that the description of the method I just offered is accurate in detail -- but it is, I have been told, correct in spirit. I tried the method, experimented with it, kept running into tactical problems, worked out modifications that seemed to suit the students and the courses I was faced with -- and I became a convert.

Some say that the only possible effect of the Moore method is to produce research mathematicians, but I don't agree. The Moore method is, I am convinced the right way to teach anything and everything. It produces students who can understand and use what they have learned. It does, to be sure, instill the research attitude in the student -- the attitude of questioning everything and wanting to learn answers actively -- but that's a good thing in every human endeavor, not only in mathematical research. There is an old Chinese proverb that I learned from Moore himself:

I hear, I forget; I see, I remember. I do, I understand.

1. Halmos, P.R.(1985). I want to be a mathematician: an automathography. New York: Springer-Verlag, p. 258.