Like in Aikido, mathematicians (and other academicians) have a tendency to take pride in their lineage. My instructor and mentor was "mathematical grandchild" of R. L. Moore, a mathematician perhaps more famous for his method of instruction than for his technical results. The Moore Method is described fairly well in the Wikipedia (link). In its essence, the Moore Method is such that the instructor creates the environment through which the student can explore and discover known material on his own as if he was the first researcher to encounter the ideas, and this with only the lightest guidance from the instructor.
To students who are focused upon the GPA and who are reduced by a spoon-feeding educational system to memorizing and regurgitating facts that will likely appear on a test, a class conducted in the Moore Method can be truly terrifying experience. To the intellectually curious, it is absolutely exhilarating!
Surviving the class means that you truly understand the material because you have created the mathematics just as your predecessors have: You followed your nose, you explored, you speculated, you pursued every lead, you found every dead end, you tested what was necessary and what was sufficient, and so forth. You have internalized the material.
Taking such a course, you also explore what it means to be a researcher, working beyond the frontier of what is known and understood. You learn self-reliance and perseverance as you walk a path, repeatedly unbalanced, struggling through plateaus of understanding.
You may also have learned what it is to to think, what it is to learn, and perhaps even what it is to be a different kind of mentor or a teacher.
It is not an easy path, and it is not a path for everyone, but it is a path of Mastery.