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I am interested in learning how a course in geometry is employed today at undergraduate colleges/universities in the U.S. On the one hand, such a course seems to serve as an optional (rarely required) upper-level course of variable content for mathematics majors. But on the other hand, often future high-school teachers need to see Hilbert-style1 planar geometry proofs. So for them, the emphasis on proofs and rigor is essential. Whereas for the upper-level math majors, the course could be differential geometry, algebraic geometry, discrete geometry, etc., according to the instructor's inclinations.

I'd be interested to learn how this tension (if I've accurately captured it) is currently resolved, and what might serve as the content of a college geometry course. Contrasts beyond the U.S. are welcome.

1 Gries D., Schneider F.B. (1993) "Hilbert-style Proofs." In: A Logical Approach to Discrete Math. Texts and Monographs in Computer Science. Springer, New York, NY. DOI link.

PS. I wanted to cite these articles by Joseph Malkevitch (@JosephMalkevitch), advocating for discrete geometry, but I am not finding public links. (Later) Author provided a link.

Malkevitch, Joseph. "The 'New Math' and Claims Discrete Mathematics is the New 'New Math'." Mathematics in School 40, no. 2 (2011): 8-10.

Malkevitch, Joseph. "Discrete Mathematics and Public Perceptions of Mathematics." Discrete Mathematics in the Schools 36 (1992). author's PDF download.

Added 21Sep2020: Thanks to Joe Malkevitch, just saw this recommendation from CUPM (MAA Committee on the Undergraduate Program in Mathematics) on Geometry: "the curricular needs in geometry of future high school mathematics teachers are best satisfied within geometry courses for mathematics majors, not by a separate Geometry for Teachers course."

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    $\begingroup$ I haven't taught it, but I've looked over some recent syllabi from a US institution. The solution was to do the HS-teacher curriculum first and finish with hyperbolic geometry for the math majors. The math majors almost certainly haven't seen it --- only heard about it --- and it can be done without introducing full-force differential geometry. $\endgroup$
    – Adam
    Sep 20 '20 at 23:21
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    $\begingroup$ @Adam Anecdotal, but in my undergraduate program in Texas, our "Foundations of Geometry" class was structured this way, except the "HS-teacher" part was more general Euclidean Geometry (directly from Books I-VI) than focused on HS standards. There was also a bridge between the two portions including more modern results in plane geometry. Hartshorne's "Geometry: Euclid and Beyond" was used as the text for the second half of the course (except all the parts requiring field theory and differential geometry). $\endgroup$
    – Andrew Sansom
    Sep 22 '20 at 23:33
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I taught a geometry course which counted both for math majors (sophomore level), and was required for the math education program secondary education teacher majors, and we used the text:

Edwin Moise: Elementary Geometry from an Advanced Standpoint, 3rd Edition,

which was amply challenging for both cohorts of students.

For math majors completing this course, I later taught a sequel course, *Differential Geometry".

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