# 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

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The above link by dato is to the book The Foundations of Geometry by David Hilbert (Grundlagen der Geometrie 1899, translated 1902 by E. J. Townsend). –  ShreevatsaR Jul 18 '12 at 14:02
–  lhf Jul 18 '12 at 14:48
Your reference to "axiomatic," "modern," and "constructive" approaches doesn't make sense to me. Euclid's axioms are constructive -- they state that certain things can be constructed. E.g., the 3rd postulate says we can construct with a straightedge. The examples you give of "the modern approach" are examples of theorems, which can be proved from Euclid's axioms. You may find something of interest here: theassayer.org/cgi-bin/… When you describe your students' inexperience with proofs -- haven't they all done proofs in high school geometry? –  Ben Crowell Jul 18 '12 at 14:51
@Ben: I intend for the "axiomatic" portion that the students give very, very rigorous proofs, adhering pretty slavishly to the axioms and not appealing to geometric intuition. Later, I want to switch emphasis, and give the kind of proofs which are typical of most other math proofs. Perhaps the difference is not as big as I imagine, but I think it is real. As for high school geometry, I don't know exactly what the students have done (I intend to find out), but I think it is these "two-column proofs", not in prose, and what they have done should be reviewed, supplemented, and extended. –  Frank Thorne Jul 18 '12 at 14:59

$\text{Dear Frank}$,

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!

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Thank you! (also wrote some comments offline) –  Frank Thorne Jul 20 '12 at 3:06

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.

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Thank you! .... –  Frank Thorne Jul 20 '12 at 3:06

For the kind of course you are talking about I believe that the historically motivated development of euclidean/affine geometry is what you could benefit the most from, being pretty elementary but deep and very insightful and motivating. I eagerly encourage you to look up this new book just released:

Ostermann; Wanner - Geometry by its History

It is full of pictures and constructions at every page, and has many hints and solutions to all problems so it would make a very useful reference for planning your own course notes/lectures, etc... It covers many of the elementary topics in depth and from an elementary level (from Thales theorem to proof of the nine-point circle and beyond).

Aside from this wonderful new book, other standard references at the mentioned level are:

Gibson - Elementary Euclidean Geometry, An Undergraduate Introduction

Aarts - Plane and Solid Geometry

Pedoe - Geometry, A Comprehensive Course

Bottema - Topics in Elementary Geometry

With those references you could develop a very interesting course, even if you have to lower the level of some books and pick up results and themes from here and there. Nevertheless, I strongly suggest you get the book by Ostermann/Wanner, because it is a gem to enjoy for any teacher and lecturer in elementary geometry. Hope these help you out a bit.

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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.

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As a high-school student, who studies Euclidean Geometry for the purpose of Mathematics Olymnpiads, I would recommend the following, not as high-powered as Coxeter, books.

The Geometry of the Triangle - Gerry Leversha

Introduction to Geometry (2 book set) - Richard Rusczyk

These are all fairly basic, but would serve as a sound introduction for books like Coxeter.

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To add my two cents, part of what's hindering and scaring a lot of people who have to teach "college geometry" these days is the utter collapse of the American high school system. As a result, it's no longer a given that your students are comfortable with what used to be "high school" geometry (something that used to be a given for any student at any university). As a result, we really have no clue where to begin, sometimes even with very talented students. These days, talented doesn't necessarily mean well prepared.

Frank, you could try the following. Robert Bittman, a top organic chemist and a spectacular teacher, used to do this and it worked great for him. Give your students a little "baseline" quiz on the first day covering the elements of high school geometry. Make sure you tell them it doesn't count so none of them hyperventilate. See where they're at in their knowledge of geometry. You can also put in some basic questions about linear algebra and/or naive set theory, to see if they know some of that as well. Then, based on the answers - or lack thereof - you can spend the next few days constructing the basic outline of your course and adjust it as you go. If you have the time, I strongly recommend writing extensive lecture notes to serve as the textbook. This will not only cement your own knowledge of the subject, but it will allow you complete control over the selection and presentation of the course material and you can make it your own. I know this simply isn't possible most of the time, but if you have the time and willingness to do it, I really recommend doing it. Best of all, you can edit and revise the notes in subsequent years if you teach the course again - which you probably will - to improve it. It's a ton of work, but it's a labor of love and the great thing about it is the bulk of the work only has to be done once. But like I said, it depends on how much time you really have for the enterprise.

Good luck!

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Something that was never emphasized to me (and I didn't figure it out until later) was why we spent so much time on proofs, etc.

I only realized later it was to show how mathematical reasoning can go from axioms to proven knowledge, with a subject matter that is intuitively grasped. I've emphasized this now as a homeschooling dad, and wish more teachers would do this.

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I personally thought knowing the difference between Euclidean and Non-Euclidean geometry was a big plus in really understanding the axiomatic approach and the idea of mathematics. Especially learning hyperbolic geometry with the Pioncare-model was so intuitive and easy to follow.

I was particularly not a smart student and I have been teaching for 7 years and learning some more math for myself, so I think I know a good book or two. I seriously recommend "Geometry: Euclid and Beyond" from Hartshorne.

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