Question 1: My first-semester calculus students have a very narrow conception of mathematics. This applies to both what mathematics is about (they think calculus is the most advanced math there is) and what its goals are (they think the most important thing is to get the answer in the back of the book). I would like my students to have a richer idea of what mathematics is about and what it means to do mathematics. It's a lot more than plugging numbers into formulas--mathematics is about ideas and the relations between them.

Question 2: I think that basics like geometry, algebra, trigonometry, functions and graphs, and statistics and probability should be part of every student's mathematical training. Let me say a few words about each.

* Geometry: This is crucial because humans are much better at imagining geometric relationships than algebraic ones. Geometry also introduces the idea of proof, which is where students learn the difference between simply accepting facts on the authority of the teacher and seeing the full reasoning for themselves. There should be more emphasis on three-dimensional geometry.

* Algebra: These days, some calculus courses place less emphasis on algebra. But I have seen otherwise capable students be crippled by an inability to do the simplest algebraic manipulations.

* Trigonometry: Much trigonometry taught in high school is useless for college. Other crucial topics, such as periodicity, are not emphasized enough. I regard an understanding of simple ideas about periodicity and frequency as "life skills".

* Functions and Graphs: Students, even those who have had precalculus or calculus in high school, have poor conceptions of what a function is. This is one place where the calculus reform movement has made a real contribution--much of the reform material on functions is first rate and tailor-made for the high school level. And being able to read a graph is another important "life skill".

* Statistics and Probability: This is another extremely important "life skill". The NSF [National Science Foundation] should fund grants in Prob/Stat Reform parallel to what it does in Calculus Reform. An understand of these topics is crucial to having a technologically and scientifically literate society.

Question 3: For college-bound students, I could tell that high school mathematics education was working well if I started seeing some significant differences in the students in my calculus classes. But for the larger group of students who don't go to college or take very little mathematics in college, I don't know what the answer is. One way to measure if high school courses in probability and statistics are working would be to see if people taking these courses spend less money on the various state lotteries!

Question 4: I think prospective high school teachers should have a good background in mathematics, preferably at least an undergraduate degree in mathematics. They need to see that there is mathematics beyond calculus and that the idea of proof is central to mathematics.

As for attitude toward mathematics, I would hope that teachers would be aware that mathematics is a living subject where new mathematics is being constantly created. I would also hope they would be deeply aware of what I mentioned in the first answer, that mathematics is about ideas and the relations between them. And, of course, one wants teachers to like mathematics and be enthusiastic about communicating it. As for pedagogical approaches toward mathematics, I don't have very strong feelings except to be wary of orthodoxies. Some students work best in group settings, while some work best individually. Many students prefer more concrete applications, while others enjoy playing with the ideas at an abstract level. I would hope that teachers would be flexible and adaptable in their pedagogy.

Question 5: What first attracted me to mathematics was the fact that I was aware from a fairly early age that I could do it. In the 8th grade in Louisiana in 1961-1962, our school tried an experimental curriculum (I think it was the SMSG [School Mathematics Study Group, also known as "New Math"]--I remember yellow paperbacks), and it was a colossal failure. Of the whole class, only two kids (me and one other) had any idea what was going on. After a month, we went back to the more traditional stuff, but my teachers and I were aware that I could do interesting mathematics.

At this time, another extremely important factor attracting me to mathematics was the influence of our local university (the University of Southwestern Louisiana). Every summer, there were math programs for talented high school students that I participated in, starting with the summer after the 8th grade. At least some of these programs were sponsored by the NSF. This is where I first learned about number theory and knot theory, for example.