Question 1: I care most of all about a student's attitude towards mathematics. I want students to know that they can always learn more, that mathematics is not a barrier to careers and hopes, but a tool to solve problems, to create opportunities, and to see and to understand beauty. If you find a student who understands these things, you have not doubt found one who is highly skilled and who knows well how to learn and to use mathematics.

Question 2: A good high school must offer a diverse mathematics program that leads toward the fundamentals of calculus, discrete mathematics, and statistics. A high school should not be measured by how many of its students take advanced placement calculus. All schools must start with the assumption that all citizens, all future voters, need to understand more mathematics than in decades past. Therefore all students need to see a higher standard of mathematics than we have had in the past. I am not advocating that all students be in the same courses, only that there be an appropriate, rich program for students at all levels of ability and background. In theory, tracking is a good idea because it can group people who can work at the same level. In practice, however, we have in too many cases evolved into programs where there is a success track, taught by strong teachers to strong students, and a failure track, taught by weaker teachers to underachieving students. We need the strong courses, obviously, but we also need to create programs that can spark the underachievers and that can position those students to take the strongest courses in their turn. We cannot have curricular dead ends. One challenge to all high schools is that, by that age level, there is a tremendous diversity in the quality and quantity of mathematics that students have seen so far. Plenty of bright and hard-working students have had uneven mathematics instruction through the middle years. It is important not to give up on those students. Please understand that I am not blaming elementary and middle school teachers for the problems of high school mathematics students--the fault is in our system as a whole--but I am only trying to point out the diversity in mathematical background one finds in many high schools.

Question 3: A high school program is working well when a high percentage of the entire population is learning new mathematics. A high number of remedial or terminal courses would not be a good sign. Enthusiasm for mathematics among a large portion of the population will be the best sign of all.

Question 4: Mathematics teachers need to be experts in their field. They need mechanical skill and conceptual understanding. They need most of all to be mathematics enthusiasts, people who always show interest in mathematics problems and concepts. As teachers they need in some way to transfer their enthusiasm to students. I believe in the model of the teacher as a coach, on the sidelines with leadership, advice, strategies and ideas, but where the students themselves really do the mathematics. On another level, teachers need to keep changing what they do and the courses they teach. There are too many teachers, I believe, who have been teaching the very same courses for a great many years. Doing so may be comfortable, but it must also be mind-deadening for both the teachers and for the students. Teachers must also have a sense of the culture and history of mathematics. I am bothered, for example, by teachers who are experts at simplifying radicals (say, changing the square root of 60 to 2 times the square root of 15) but who cannot explain why it was once important to do so. Teachers need perspective so they can better judge what is really important to know.

Question 5: I grew up in a family full of engineers, scientists, and mathematicians, along with a few artists and writers. I have a university background in mathematics, but I also studied English literature. I have settled into mathematics because of its diversity, because of fascinating new ideas in the field from dynamical systems to discrete mathematics, and because I like to solve and discuss mathematics problems. And although I am not a research mathematician, in my own practical way I am a research mathematics teacher. I suspect that there is a better way to teach most ideas, and I enjoy trying to develop those ideas. The challenges of technology and an unpredictable future only make mathematics teaching a more interesting challenge.

Question 6: a) My hopes for higher education in mathematics echo my hopes for high school. We should continue to offer a rich, intense program for those who wish to specialize in mathematics, but there should also be an expanded effort to educate a more general population about the power, beauty, utility, and the culture of mathematics. Doing so would help reduce the isolation of the specialists , would increase the intelligent use of mathematics among all citizens, and would give mathematics the cultural impact it should have. On another level, colleges must try to accommodate students who have had a poor background in mathematics through high school. That population, a large one full of untrained talent, is one that universities can only ignore at their own peril. In a sense, this writing exercise is part of what I want to see. I know that I have more to learn about mathematics and about teaching mathematics, and I suspect that the same is true for university professors charged with teaching. Good lines of communication about content and pedagogy can only help at all levels.

b) Prospective mathematics teachers should have a strong background in classical and modern mathematics. But there should also be a rich discussion of how people learn and new ways to teach. Communication skills are vital, and would encourage all mathematics teachers to work on writing and speaking.