Some thoughts from Mark Palko:
Of all the subjects a student is likely to encounter after elementary school, mathematics is by far the easiest to teach yourself. . . .
What is it that makes math teachers so expendable? . . .
At some point all disciplines require the transition from passive to active and that transition can be challenging. In courses like high school history and science, the emphasis on passively acquiring knowledge (yes, I realize that students write essays in history classes and apply formulas in science classes but that represents a relatively small portion of their time and, more importantly, the work those students do is fundamentally different from the day-to-day work done by historians and scientists). By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians. Unlike music and writing, however, mathematics starts out as a convergent process. . . .
This unique position of mathematics allows for any number of easy and effective self-study techniques. . . . All you need is a textbook and a few sheets of scratch paper. You cover everything below the paragraph you’re reading with the sheet of paper. When you get to an example, leave the solution covered and try the problem. After you’ve finished check your work. If you got it right you continue working your way through the section. If you got it wrong, you have a few choices. . . .
I have nothing to add to this except to agree that, yes, doing mathematical research (or, at least, doing mathematics as part of statistical research) really is like doing math homework problems! An oft-stated distinction is that homeworks almost always have a clear correct answer, whereas research is open-ended. But, actually, when I do math for my research, it surprisingly often does work out. Doing (applied) mathematical research is a little bit like waking through the woods: sometimes I get stuck and have to work around an obstacle, and I usually don’t end up exactly where I intend to go, but I usually make some progress. And in many cases the math is smarter than I am, in the sense that, through mathematical analysis, I’m able to find a correct answer that is surprising, until I realize how truly right it is.
Also relevant is Dick De Veaux’s remark that math is like music, statistics is like literature.