Question 1: I want my students to understand that mathematics is widely applicable, quite beautiful, and very much a work in progress. I want them to recognize that creativity plays a role, and that the connections between models are as important as the models themselves. I want them to know how to use technology to explore, logic to verify, and language to explain their ideas. I also want them to believe that mathematics is accessible. I am much less concerned than I used to be about computational skills and about the necessity of having them before doing anything significant.

Question 2: I think that every high school curriculum should follow the four-year precalculus sequence, but with less emphasis on computational subtleties and more emphasis on modeling, problem-solving, connections, and applications of technology. The algebra and the geometry should probably be integrated, but not before teachers and textbooks are ready. A healthy majority of students should begin the sequence in 8th grade and take AP Calculus as seniors. There should be more emphasis on statistics, numerical data, and matrices, exploiting the abilities of new, inexpensive machines to handle these easily. Students should see computers often, but they themselves should use graphing calculators to do mathematics. (Schools can not keep up with computer technology today, and it is naive to expect that they ever will. Graphing calculators, on the other hand, are both affordable and effective. The computer distraction is effectively SLOWING DOWN the incorporation of technology in many schools!)

Question 3: The best way to determine how well high school mathematics education is working is to develop meaningful assessments that actually test what we value. Not many of the current national assessments do that, but the testing folks -- driven by the marketplace -- are scrambling to develop some. Meanwhile, we can expect plenty of conflicting anecdotal evidence while college professors try to apply old expectations to students with new mathematical talents. When the high school reforms and the calculus reforms begin to mesh, we ought to see more mathematics majors, better mathematics students, and a workforce that is better prepared mathematically.

Question 4: High school teachers need first of all to want to reach all their students. It may sound corny, but all sorts of problems arise when this prerequisite is either lacking to begin with or else sublimated to achieve other goals. Secondly, they need a broad mathematical background that will enable them to guide their students through wide seas rather than narrow channels. Thirdly, they must appreciate, use, and do mathematics in their own lives so that they can inspire their students to do the same. Fourthly, they must be willing to look beyond their own mathematical upbringing to find more effective ways of getting their students to learn mathematics (such as group learning, student discovery, and alternate assessments). Finally, for sanity's sake, they must understand children well enough to rise above their own emotions when dealing with those of their students.

Question 5: I decided to enter into mathematics because I was good at it and I thought that I could teach it. Although I have a PhD in Combinatorics, I decided to teach at the high school level because the students would be more interesting and the commitment more varied. Since I became a high school teacher, the rewards have been such that I have never really looked back.

Question 6: With apologies to my AMS colleagues, my expectations for my students as they go on to colleges and universities during this time of transition between traditional and reformed courses are not optimistic. Those who are fortunate enough to have teachers who welcome technology, value conceptual understanding, and encourage collaboration will do well. Others will turn to other departments and do well, and ironically will use their mathematics -- because using mathematics is what I have been encouraging them to do all along. The creative thinkers, the ones whom I have tried most earnestly to woo, are the ones most likely to leave. The ones who are most likely to stay are the ones who believe that mathematics is a set of examples to be mimicked for credit.

Having said all that, let me paradoxically aver that I believe the prospects for future mathematics TEACHERS to be very bright indeed. The calculus reform movement has spawned a major refocusing on undergraduate education, which has turned the attention of some of the most creative minds in our field (and certainly our best teachers) toward teaching. It is the students of THESE teachers who will be the teachers of tomorrow, and mathematics education will prosper through natural selection.