That's one reason I liked my non-science/math courses in school better than the science/math ones, despite otherwise being inclined in the other direction. It varied based on the course, but on average I felt that courses where you wrote a paper or essay took the problem-centric approach more seriously, and graded me based on whether my analysis was interesting, coherent, supported properly, took into account obvious objections, etc., rather than by whether it was the "right" analysis matching some pre-determined answer the teacher was looking for.

I did have a few project-based science courses, based more on having to propose something and justify your proposal, rather than giving a pre-determined answer. But I wasn't able to find more than a handful of those, and they tended to only be senior-level elective seminars, with a small number of students. I would guess the trend towards large, lecture-based courses (whether in-person, or MOOCs) will make them even more rare.