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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 ?

Likewise,

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.

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    $\begingroup$ I'd be curious in what motivated this question at this time...? $\endgroup$ – Daniel R. Collins Dec 6 '16 at 17:15
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    $\begingroup$ Do you consider a student who drops the course halfway through as not passing the class? Or are you only referring to students who receive failing grades? In my university, the students perceive a failing grade as potentially life-altering, so most students who are likely to fail the course will drop the course in the middle of the semester (right before my university's deadline to drop courses). $\endgroup$ – Michael Joyce Dec 6 '16 at 18:55
  • $\begingroup$ @Michael Joyce Yes, we can include withdraws as not passing. The D-W-F rate is what is most relevant. $\endgroup$ – James S. Cook Dec 6 '16 at 21:15
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    $\begingroup$ In contrast, electives have better pass rates because, well, students elect to take them. This is not necessarily accurate in general. You were a math major. There is some nice info on this in ch. 7 of Valen Johnson, Grade Inflation: A Crisis in College Education. Data show that, e.g., chem majors do better in drama classes than they do in chem, while drama majors also do better in drama classes than they do in chem. There are simply massive differences in grading standards between the liberal arts and STEM. $\endgroup$ – Ben Crowell Dec 8 '16 at 23:13
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    $\begingroup$ @BenCrowell no doubt. It's interesting that a study has been done to show that. I think we (as in us math or physics folks) well remember that classes outside our major were usually easier. On the flip side, this is certainly part of what teaching math or physics such a comparatively troublesome enterprise. That is, if our "success" is judged by the satisfaction of the students. Any gen-ed course we teach has the deck stacked against it. Every semester I read course evaluations about how much harder my course was relative to everything else. $\endgroup$ – James S. Cook Dec 9 '16 at 0:11
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My experience has been that the grade distribution curve in upper-level classes is different, and, I think, rightly so.

The incapable/uncaring students were almost certainly "culled" by an earlier class. Viz. If students come into the school on a gaussian of "ability" (univariate for simplicity, this generalizes to many dimensions), then as time goes by the left tail gets chopped off, and by the time students are in abstract algebra, you only have the right tail of students, and it is reasonable for their grade distribution to be skewed right... On the other hand, if we assign a gaussian of grades for every course, and the bottom x% fail out of principle, then after y courses, only (1-x)^y% remain. If x is 10%, and y is 12 courses, that is 28% of students. If your school only graduates 28% of the students it accepts then it needs to reevaluate its acceptance criterion.

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    $\begingroup$ Actually, there exist a sizable fraction of comparatively uninterested students in higher courses. The early classes do not cut out that portion of the students. Partly this is due to the front-loading of calculus I, II and III within the major. That 12 hours is largely free of proofs. In contrast, the fraction of the course which is proofs continues to grow and culminates in the real analysis and abstract algebra course. ... $\endgroup$ – James S. Cook Dec 10 '16 at 15:04
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    $\begingroup$ ...So, for the student who is computationally marginal and proof-wise inadequate this fractional split in focus (computation vs. proof) exacts a heavy price in these final courses. If I understand your answer correctly, perhaps we should start students with the proof course to help those who are unable to pick a different major. But, I doubt I can sell such a proposal here as the major is already small and in an environment which so strongly frowns upon small majors, we hardly want to be smaller. $\endgroup$ – James S. Cook Dec 10 '16 at 15:05
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    $\begingroup$ I was a philosophy double major, and had several logic classes before my mathematics classes started being almost exclusively rigorous proof. I would agree that early proof-writing was handy. $\endgroup$ – Scott Dec 10 '16 at 15:12
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    $\begingroup$ I think the math major is useful in the liberal arts sense of creating well-rounded problem solvers. If you can understand the theory and structure of real analysis and abstract algebra then that lets loose a part of your brain which is hard to unlock by less vigorous mental exercise. This is even more true for physics majors. I believe these majors prepare students for many jobs which have little to no direct connection with math or physics... $\endgroup$ – James S. Cook Dec 10 '16 at 15:32
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    $\begingroup$ I greatly sympathize with these dilemmas. Our community-college does in fact currently graduate just about 28% of the students who enter (giving it one of the highest graduation rates in the U.S.) Revisiting acceptance is not an option, since we are open admissions. Instead, our institution is currently requiring that we massively reduce both gen-ed and STEM major requirements. $\endgroup$ – Daniel R. Collins Dec 10 '16 at 23:33

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