Devlin’s Angle recently published a nice synopsis of the work of R. L. Moore, one of the great mathematics instructors of the 20th century: http://devlinsangle.blogspot.com/2015/02/the-greatest-math-teacher-ever.html

Moore’s method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture — and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt — or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student’s presentation.

Very often, the first student would be unable to provide a satisfactory answer — or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem.

Moore’s discovery method was not designed for — and probably will not work in — a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher — and possibly the right students — Moore’s procedure can work just fine. Moore’s own area of general topology is just such an area. You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they rarely meet with the same degree of success.

Lest this sound extreme, here’s a snapshot of Moore’s educational philosophy produced:

If you measure teaching quality in terms of the product – the successful students – our man has no competition for the title of the greatest ever math teacher. During his 64 year career as a professor of mathematics, he supervised fifty successful doctoral students. Of those fifty Ph.D.’s, three went on to become presidents of the AMS – a position our man himself held at one point – and three others vice-presidents, and five became presidents of the MAA. Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own and helping shape the development of American mathematics as it rose to its present-day position of world dominance.

Present-day inquiry-based learning uses some (but not all) elements of Moore’s teaching style. I personally do not teach using the Moore method, as I have developed my own teaching style that seems to work well with my students. That said, I would seriously consider using the Moore method for classes which are best suited for this style of pedagogy.