Question 1: I care most that they understand algebra and that when they make symbolic manipulations e.g., to simplify expressions, they understand not just what they are doing (the mechanics), but why that is what should be done. I frequently see "knowledge of how to do a problem that is exactly like a problem seen before". I hope that we can teach them how to apply principles to solve problems which are not just like ones they have seen before. But to apply the principles, they need very good command of the algebraic manipulations.

Question 2: That students come away with a good understanding of the mechanics and meaning of Algebra II. (I assume that this cannot be done without understanding Algebra I.)

Question 3: From my experience of teaching calculus, I would say when students arrive knowing the things I mentioned in 1 and 2 and, particularly if they have had calculus, knowing what they understand and what they know. A lot of students arrive thinking that because they had a calculus class in high school, they know everything about calculus.

Question 4: I believe that a variety of pedagogical approaches can be successful. I suspect that the most important thing is the attitude the teachers bring to mathematics. I know my son suffered from number of teachers who insisted that mathematics was about right answers and that there was only one way to do a mathematics problem correctly. I think there is a big difference between "only one correct answer" and "only one way to get the answer". My feeling for a strong math background for teachers is based on my belief that the better the math background, the more flexible the teacher is likely to be about mathematical work.

Question 5: Arithmetic. I started to do well in mathematics when it was arithmetic and since I continued to do well in high school, I assumed that mathematics was the major for me. If I had known what mathematics was, I might have chosen another major.