I am curious, if we define success or failure by grades earned, or pass-rate (pass meaning the student does not need to retake the course) then what is the typical pass rate for students in upper level courses at universities in the US. Of course, this question seems too broad, hence let us refine the question further:
For institutions which have a required course in abstract algebra (at the level of Gallian, Fraleigh etc. ) what percentage of students do not pass the class ?
For institutions which have a required course in real analysis (at the level* of Pfaffenberger, Lay, Krantz etc. ) what percentage of students do not pass the class ?
Failing either of these classes implies the student is not ready to graduate with a degree in Mathematics. It follows the students either need to retake the course, or, seek a transfer of these courses from a different institution.
Let me offer my own personal anecdote here. When I was an undergraduate in Math and Physics I always saw some subset of my peers fail the courses I was required to take. In retrospect, I think comparatively high fail rates are symptomatic of the course being required as to complete a degree. In contrast, electives have better pass rates because, well, students elect to take them. Getting back to my point, I certainly thought there was danger of getting a bad grade in my core math or physics courses, especially at the junior or senior level. So, as an instructor, the idea that some students don't pass the required courses I teach is not particularly surprising. This brings me to my third and related question which probably should be its own, but, I include it here to complete my thought.
Is it unreasonable to expect students to retake a major course in their last year ? How should we deal with the students who don't succeed in the higher level courses?
I appreciate both anecdotes and links to data. Thanks!
*admittedly, I am not sure these texts are on the same level, perhaps I should include baby Rudin in this list, anyway, I'm trying to describe what is usually a 400-level, often terminal, course on real analysis for math majors.