Suggestions for teaching advanced high school students Hi all,
I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don't quite know where they're at just yet. I assume they've done Euclidean geometry, analytic algebra are just getting used to differential calculus. I'm fairly free to do what I want with them short of cruel and unusual punishment (such as making them read Gilbarg and Trudinger). They meet weekly and I had in mind meeting them every two weeks and going through a shotgun summary of what maths is all about, with an eye for a historical development. Of course, I want it to be fun! I have a semseter, so that makes about 6 or 7 two hour meetings. 
To make the question precise: What do the potential answerers believe should not be left out of such an introduction?
Thanks, 
Mat
 A: Trying to cover the material in a course is usually bad. You don't have much time, and even if the students learn the material, they might just be bored when they see the material later in a normal class.
I think you have an opportunity to let the students do some mathematics, and to see some of its beauty and challenge. Don't make the mistake of trying to get them to work on open problems, but do give them activities related to the following parts of mathematics:


*

*Guess what is true. 

*Prove what you believe is true. 

*Communicate.


It was a big surprise to me when I learned in the PROMYS program that these are what mathematicians do as opposed to the types of things I had seen in mathematics classes, which concentrated on learning techniques and applying them. I do not recommend trying to follow PROMYS because you have much less time.
For example, you can start with Pascal's triangle. Have them look for patterns. Suggest looking at the even-odd pattern, or the sums of every element in a row, every other element  or every third. Estimate the size of $2n \choose n$, and ask how you might compute ${30 \choose 15} \approx 2^{27}$ with $32$-bit numbers so that you can't compute $15! \approx 2^{40}$ directly. (Recursion? ${30 \choose i}/{30 \choose i-1}$? Prime factorization?) Ask for generalizations to multinomial coefficients. Show some connections to other areas of mathematics, e.g., ask how many faces a hypercube has of each dimension, or mention the Central Limit Theorem. You can show how you might prove some of these patterns with induction, or bijective arguments, or by evaluating $(x+y)^n$ at particular values of $x$ and $y$. Have them read an article covering related material in something like the College Journal of Mathematics or Quantum so that they see some good exposition, and that mathematics is still active. Let them flesh out and present parts of the article. Then have them write something about what they have learned.
An advantage over some competitions which emphasize distinguishing the most exceptional individuals is that there is something for everyone in Pascal's Triangle (or in the exploration of similar rich subjects of mathematical study). If your students have a lot of talent and energy, they can work on complicated material. If they are not as advanced, then they can still make real progress on simpler patterns.
An historical overview in parallel might complement the hands-on study. 
A: The site mentioned in Greg's answer, Art of Problem Solving, was my old haunt in high school (the pre-MO days!) so perhaps I should say a word about it. First, there is a large forum there with lots of people posting interesting questions, and you might recommend your students to browse and perhaps join; that was my main source of practice at doing mathematics in high school. Second, there is a Resources page with previous questions from many famous Olympiad-style competitions; I'm really only familiar with the US competitions and the IMO, but of those I can suggest you look at USAMTS and AMC 10 questions for interesting problem sets. 
Of course you are not training these kids for an Olympiad so the focus on Olympiad material shouldn't go too far. Nevertheless, I agree with Greg that doing mathematics is much more engaging than listening to it, so I think you should be devoting a decent amount of your time to finding interesting problems for them to do, structured around certain general topics if possible. The Art of Problem Solving books, although geared somewhat to US students preparing for competitions, represent what I think is a pretty good selection of interesting but elementary material. 
(This is all assuming they aren't already hardened Olympiad veterans.) 
A: Show them the rational parametrization of the unit circle. Then generalize the method to other conics with an obvious rational point like $x^2 + y^2 = 5$ and $x^2 - 2y^2 = 1$ and discuss why it doesn't work on $x^2 + y^2 = 3$ (i.e., how would you show this conic has no rational points at all). Finally, discuss the parametrization of the integral points on $x^2 - 2y^2 = 1$, where the story is quite different from the case of rational points. 
A: Most likely, there is an established culture of mathematical circles in your country or even city; you should get in touch with your local "branch" for ideas about the scope and curriculum for your course. You might want to include your country/city in your question to get more localized advice.
There are many online resources, too, like Art of Problem Solving.
Problem solving and olympiad problems usually works very well, while teaching a university level course on, say, complex analysis early usually doesn't. The mathematics should be elementary and easily accessible (good: elementary number theory, graphs, polynomials in one variable, combinatorics, Euclidean geometry; bad: differential calculus, advanced algebra) and the focus should be on the students doing mathematics instead of listening to it.
A: The most valuable thing that you have to offer these students is your own experience as a college student and graduate student. This might seem surprising, but is based on my own experience of working with high school students in the U.S.  Many of these students aren't sure about going to college, don't know what will be expected of them when they do go to college, and aren't sure that they're "smart enough" to succeed once they get to college.  Mostly they need mentoring and encouragement more than they need instruction in any particular aspect of mathematics.  
There's a good chance that you know more about what's going on in college than even their high school teachers (who may have been out of college for many years.)   You're closer to their own age and not an authority figure, so they're more likely to believe what you have to say.  You should talk to them to find out what would be most useful for them to know.    
A: Things that seem trivial or stale or just very familiar to you as someone who thinks about mathematics every day will be new and fresh and exciting to kids who have never seen them before.  Examples include symmetry groups, Euler's formula and the Argand plane, the quaternions (and rotations, and the soup bowl trick), continued fractions, infinitesimals (don't underestimate the the thrill of "breaking the rules"), the logistic map and chaos...  If you're not sure exactly what to do, don't put all your eggs in one basket by planning a rigidly structured programme.  Just show them a bunch of cool stuff, and if something really catches their imagination you can take it further.
A: Assuming that you are dealing with U.S. high school students I could say something. At Augusta State University I am in charge with AMC 12/10 and Putnam (which is indeed Collegian competition but I alway have one or two very talented high school kids). I put two websites, one for AMC 12/10 ( http://predrag.freeshell.org/AMC/amc_about.html ) and one for Putnam ( http://predrag.freeshell.org/Putnam/putnam_news.html ) 
Carefully open all links on the bottom page with resources. My favorite however (probably I am nostalgic a bit) is link to Kvant ( http://kvant.mirror1.mccme.ru/ ). Serge Tabachnikov of Penn State have published two books of translations of most memorable articles. 
I also like to use Gelfand's high school text books 
For example Algebra 
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
I like very much articles from three volume translation of 
Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov and M. A. Lavrent'ev. 
http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=mathematics+its+meaning+contectst&x=0&y=0
However, the articles are very, very challenging.
There is also a very famous Encyclopedia of Mathematics for high school students in Russia but I am not sure if it is translated into English.
Look for Vinogradov's book on number theory. It very, very deep but accessible for high school kids. 
A: I would recommend actively involving them by giving them some problems to investigate and try to solve (and choose things they could actually attack) ... These could be like all the gems buried in Pascal's triangle mentioned above, some basic number theoretic things, some geometry deeper and more interesting than typical high school geometry. I don't recommend showing them advanced mathematics that requires college or graduate background, or even calculus, at this stage.
A: You may get a lot of suggestions to keep the course fun and entertaining; here is one to keep it serious and down-to-earth.  Fortunately, the essence can be presented in a few minutes, and a good job on it need take less than half an hour.  If they want to communicate their results professionally, they will need to learn platform skills and writing skills, but they will need research skills even more.  Some suggestions as to how to find references, evaluate their credibility, and cite them in reference to their own work, should come before learning how to find a journal or conference (or even another mathematician) to submit their results. It may be fun to reinvent the wheel in your own way; I suggest it is more satisfying to use it and other pieces to build and sell a car.
Gerhard "Email Me About System Design" Paseman, 2011.06.26 
