I'm teaching a college geometry course. What should I cover?

I've been asked to teach "Foundations of Geometry" at the University of South Carolina. Apparently, professors in the past have all done very different things, and I have a lot of choice in the matter. What course would you like to see?

Some constraints:

• I want to say at least a little something about (1) the axiomatic approach (e.g., Euclid's axioms), (2) the modern approach (cool theorems such as Ceva's theorem, the nine point circle, etc., etc.), (3) the constructive approach, with straightedge and compass.
• My students differ in background and motivation. Most of them are prospective high school math teachers. Typically they have seen some proofs, but not a lot; for example, they may have proved that the sum of two odd numbers is even. Most of the students won't yet have taken analysis, or algebra, or any other course obliging them to work really hard.
• It has been a long time since I have seriously dealt with the subject, so I will definitely want a good book (or books) or other materials to follow.
• I won't say anything about projective or other non-Euclidean geometry, because there's another course for that.

I asked a similar question on MathOverflow, but I was a little bit spooked by the answers. I'm sure that the books of Hartshorne and Hadamard are excellent, but I suspect these might be better for stronger students. (And the Hartshorne book discusses "geometry over fields", etc., and most of my students won't have taken abstract algebra.) Some respondents rolled their own solutions -- but are there really not good source materials out there?

My colleague taught a course out of Isaacs' book -- but he really went the extra mile (more like the extra ten miles). He said that the exercises were a bit too difficult, and that he was constantly having to give the students hints, and he slaved over writing up complete solutions. I believe this would not work as well for me as it would for him: he has a very approachable demeanor that I haven't (yet) been able to duplicate, and I'm afraid the students would be unlikely to come to my office hours no matter how much I encouraged them.

I've also looked at other books -- Coxeter and Greitzer is very cool, but perhaps more naturally suited to hotshot high school kids. Posamentier looks promising, a review copy is on its way. Clark's book seems to be somewhat off the beaten path (where I'm not quite sure I want to follow), but it looks very interesting and I ordered a review copy of that also. Other suggestions, big or small, would be very welcome. Thank you so much! --Frank

$\text{Dear Frank}$,

Hi there. I have never taught the exact course you're describing, so the answer I'm about to give is almost entirely about the experience of teaching mathematics to the kind of clientele you're describing.

First of all, I think that when planning and calibrating the course, by far the most useful information will be information about the students who have taken it in the recent past. From what I have seen, students in math education programs have a high degree of "program loyalty": they have taken a lot of courses with similar (not tremendously high) expectations and protocols, and they are often unpleasantly surprised at the way "real math classes" run. Of course this does not mean that one should just accede to their lower expectations (and the better students won't really want that, but they may be more culturally averse to voicing those feelings than we're used to from math students), but these perceptions are something to be kept in mind when designing and running a course.

Let me speak from my own experience as someone who has to work hard to "lower the intensity" in most undergraduate courses I teach. One thing that I try to do when I teach a course for the first time (or even for the first time at a new institution) is to make sure that my course will look comfortably similar to previous incarnations of that course taught by other people. So for instance in teaching an undergraduate course on introduction to proofs, rather than look through the very wide offering of intro-to-proof books of various shapes and sizes, I just asked "What text do they usually use for the course?" I was told there was no standard text, so I asked "What text did they use last year?" And they told me that two people that I knew to be reliable teachers had used a certain text. I thumbed through it to make sure I could live with it, and I went with that text. And then, when teaching the course, I for the most part just chilled out and followed the text, remembering to periodically ask around how long other instructors had spent covering various topics. Luckily the text really was pretty good; I taught the same course again the following semester and was plenty happy to use the same book and generally try to pitch down the middle of the plate the same as the first time.

In your case, it seems that you have already found out that the last textbook used is not as cohort-friendly as you want. So my extremely unimaginative advice is: just go back a little farther. Ask what books were used for the last five years, and talk to people who have taught the course multiple times and that you believe to be successful, conventional teachers. If in doubt, I would err on the elementary side: students at this level are not good at compartmentalizing the 10% of the text that goes beyond their background and concentrating on the other 90%. If the text really turns out to be too easy, that's not such a bad problem to have, and you can supplement it with more stuff much more easily than you can cross out all the things in a text like Hartshorne which will freak your students out.

Two final comments on what you wrote:

"It has been a long time since I have seriously dealt with the subject, so I will definitely want a good book (or books) or other materials to follow."

I'm not sure that a course like this calls you to seriously deal with the subject. It is natural to want to really book up on things before you teach a course, but again, trying to stick for the most part with what's in the text is a good "cooling off" strategy for a course like this.

"the modern approach (cool theorems such as Ceva's theorem...)"

According to this wikipedia article, Ceva's Theorem is actually more than a thousand years old!

While not specific to the geometry you will be teaching, I recommend this.

When I taught maths to final school year students in the UK, near the end of the class I always did (or attempted) a relevant problem they chose, from the text book, that I had not tried before -- I was honest about this. I tried to solve it there and then to show them that the slick, logical, sequential proofs and solutions they had been shown, or read, were really the edited result of a messy human process, lots of head scratching, crossings out and dead ends. This went down well with the classes, and was highly commended by the schools inspector.

Needless to say if I couldn't do it there and then I always could by the next week's class -- I had to.

I'm in a similar position - assigned to teach an undergraduate course in geometry, but in my case, the audience will be exclusively prospective maths teachers (and it's a new course). I've been poking around a little - haven't got down to it seriously yet - but two books I'd recommend are Geometry - A Guide for Teachers by Sally and Sally and the CBMS volume the Mathematical Education of Teachers. The latter has general advice about the kind of things prospective maths teachers need to know about different areas of maths - including, of course, geometry.

An important issue for me is to be aware of the version of synthetic Euclidean geometry that is on the national secondary school syllabus (there's a uniform state syllabus here in Ireland). I don't know if this is prescribed by NCTM (or state) standards in the US. I plan to teach this by looking at slightly different choices of defined and undefined terms and of axioms etc, to see how an axiom in one system can be a theorem in another. A local mathematician wrote an interesting note (see p.21) on this topic: this has strongly influenced our recently revised secondary school geometry syllabus.