I disagree that math is easy to self-teach: The fundamental things being taught to math majors are how to distinguish between valid and invalid reasoning, and

I disagree that math is easy to self-teach: The fundamental things being taught to math majors are how to distinguish between valid and invalid reasoning, and how to produce arguments (proofs) that contain only valid reasoning. For better or worse, the way these skills are taught is to pick a couple of subjects (like analysis, algebra, or number theory) and demand that students produce dozens of proofs, which are then mercilessly picked apart by professors and grad students. That negative reinforcement is combined with lectures that (should) consist entirely of good examples of mathematical reasoning, at that's an Education. (The fact that students may also pick up a few facts about analysis or algebra is just a bonus, it's not the main point.) You can certainly get the good examples from books and video lectures, but not the experience of having your arguments criticized by experts. This part is crucial -- it's just as easy to miss the flaws in your own proofs, especially your early attempts, as it is to miss the flaws in your own programs. (Worse, it's perfectly possible to reach a conclusion that's correct through an invalid argument. You're wrong, but you'll never be able to find evidence against your conclusion.) Feedback from experts is essential.