Paul Halmos describes the Moore method
Paul Halmos described the teaching methods of by Robert Lee Moore in his book I want to be a mathematician (Springer, 1985). We give an extract below:
Moore felt the excitement of mathematical discovery and he understood the relation between that and the precision of mathematical expression. He could communicate his feeling and his understanding to his students, but he seemed not to now or care about the beauty, the architecture, and the elegance of mathematics and of mathematical writing. Most of his students inherited his failings as well as his virtues (diluted, of course); only the greatest, such as Wilder and Bing, could overcome the handicap of being a Moore student and become genuine mathematicians.
He was a Texan, almost the fictional prototype. He spoke Texan, he was politically rigid, he had strong prejudices, he stood up when a lady entered the room, and (the story goes) he wouldn't accept students who were black, or female, or foreign, or Jewish. (At least a part of that is false: he had women Ph.D. students, notably Mary Ellen Estill Rudin. So far as I know he did not have a black student.)
Moore, the educated well-spoken Texan mathematician extraordinary, was a hero of mine; Moore, the mathematically outmoded and ethnically prejudiced reactionary power, was a villain.
An effective hero, a productive villain, everyone must admit. He turned out a record-breaking number of Ph.D.'s in mathematics; they loved him and imitated him as far as they could. He did it by what has come to be called the Moore method. It is also called the Texas or Socratic or discovery or do-it-yourself method.
At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: "Mr Smith, please prove Theorem 1. Oh, you can't? Very well, Mr Jones, you? No? Mr Robinson? No? Well, let's skip Theorem 1 and come back to it later. How about Theorem 2, Mr Smith?" Someone almost always could do something. If not, class dismissed. It didn't take the class long to discover that Moore really meant it, and presently the students would be proving theorems and watching the proofs of others with the eyes of eagles. 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!