Question 1: I find this question a bit simplistic because the learning environment for teenagers today is so different, and often troubled, from the one I grew up in. For example, in some schools, my highest concern--before anything mathematical--would be that the students don't get hurt or killed in any of their classes. In many schools, one must care that antagonistic students don't undermine the basic rudiments of learning in the classroom. In many other schools, one must worry about having teachers in high school mathematics classes who have some training in mathematics.

Now if I want to be idealistic, then I would ask for the students having an exciting, motivating teacher who got the students engaged in--caring about--mathematics for a year. I do not really care about the content. I want students who have had experience exploring mathematics, asking mathematical questions. Geroge Polya always said that asking questions is the hard part; with the right sequence of questions, the answers are easy. I also want the students to acquire some study works (a student with good habits is in the top 20%, probably higher, in terms of future prospects in the study of collegiate mathematics).

I should note that I find the general thrust of the NCTM Standards [National Council of Teachers of Mathematics] very supportive of the high school mathematics experience I want students to have.

Question 2: There are core topics in algebra that every student should know. Beyond that, I am at a bit of loss. The process of rigorous mathematical thinking and exploration can be undertaken in many ways. I tend to believe that the mathematical learning process is more important than spcific content learned. I think that the NCTM Standards is a reasonable beginning for one version of a good high school mathematics curriculum, although it needs to go further for more talented students.

Question 3: Such success with students would be more instinctively felt than quantitatively measured, although of course tests would measure their increased mathematical skills.

My father loved to tell the story of his initial meeting with John Milnor who happened to be one of his freshman advisees and how impressed he was with the way Milnor answered his questions. As they discussed what mathematics Milnor had learned on his own in high school, my father clearly saw that he was talking with a student who had seen and mastered a lot of mathematics. But more importantly, he remembers feeling more and more like he was talking to a colleague, not a student.

Question 4: Items #1 and #2 cover this.

Question 5: I came from a mathematical family and when I was 3 years old, I "knew" I loved mathematics and wanted to get a PhD in mathematics.