TITLE: Suggestions for teaching advanced high school students
QUESTION [5 upvotes]: 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

REPLY [4 votes]: 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.