Question 1: I think that all high school students, whether college intending or not, should develop enough mathematical knowledge in order to make intelligent decisions based on sound mathematical reasoning. This means that all students should have sound arithmetic, statistical, and probabilistic skills so that they can create, interpret, and analyze the diverse data they will encounter in their lives. Students should have enough algebraic skills that they can use the techniques of algebra to mathematize and analyze situations they may encounter. In addition, students should be facile in both spreadsheets and other forms of technology and be able to use these tools to help in obtaining solutions to mathematical questions they may encounter. Hopefully all students will also have enough mathematics in order to know that mathematics is a unique human discipline, worthy of further exploration in their lives.

For the college-intending student, I believe that students should have a deeper understanding of mathematics. They should understand the role of proof in mathematics and have been exposed to it and know that it is the essential item that separates mathematics from the other sciences. These students should know that mathematics goes beyond just its utilitarian nature and is a body of knowledge worthy of studying for its own right. College-intending students should have a thorough understanding of function and its unifying role throughout the mathematics curriculum. These students should have a firm foundational background in algebra, geometry, probability, statistics, and trigonometry. Hopefully, through the use of technology, they also have a rudimentary understanding of the concepts of calculus, especially in the area of optimization. I would hope that these students would also have been exposed to a variety of discrete mathematical topics.

Question 2: I believe that in order to be a viable member of society, students need a strong mathematics curriculum. At the root of this curriculum is a strong content base. I would not be willing to water down current curriculum, but think, if anything, it should be strengthened. A strong content base is essential in order to make meaningful decisions based on sound mathematical reasoning. Exactly what this content is I am currently not sure, since technology has radically changed the world in which we live, and I fear that our high school curriculum has not changed sufficiently in light of these external changes. Whatever it is, it should be demanding, intellectually challenging and fulfilling, and provide our students with the tools necessary to make informed decisions. As I mentioned in Question 1, I think this curriculum should contain a strong foundation in algebra, geometry, trigonometry, probability, statistics, and discrete mathematical topics. The curriculum should expose and encourage students to use technology as a problem-solving tool.

In providing this strong content-based curriculum students need to be given opportunities to apply this knowledge in a variety of realistic and interesting settings.

Question 3: This is perhaps the most difficult question in the list. I do not know how to answer this. I believe that no single assessment device can capture this. Definitely what is deemed successful by a college mathematics department trying to assess if an incoming prospective mathematics major has an adequate background may be vastly different from what a prospective employer of a high school graduate would deem successful for their company. Certain items are easier to test than others. Basic mathematical knowledge is fairly easy to test. Trying to assess a graduate's ability to do higher-level mathematical thinking or to be able to apply mathematics in a new and unfamiliar situation is very hard indeed.

When both employers of high school graduates, college academic staffs, and employers of college graduates are not satisfied with the mathematical knowledge of these students, then it is clear that the high school mathematics curriculum is not working well. Other than that, I think it is very hard to assess this in any quantitative manner.

Question 4: You cannot divorce these ideas. Number 1 is the total package of content to which our students are exposed. If there is no exposure, there is no possibility of the student acquiring the necessary mathematical knowledge. Beyond this though comes the very important ideas of attitude and pedagogy. All the content thrown at students is worthless if the students believe this content is not important for them to know. All this content is also worthless if students are not encouraged by teachers to reflect on these ideas, generalize mathematical ideas, use these ideas in meaningful ways, and begin to see how they as students are capable of doing mathematics. Pedagogy is very important to the learning process and cannot be divorced from the content of mathematics. Again, though, pedagogy devoid of meaningful mathematics is also worthless. It is the interplay of a strong content base presented in an interesting and meaningful manner that creates the strongest curriculum possible.

Question 5: A combination of things. A couple of exciting mathematics teachers both at the high school and college level. An infatuation with the mathematics itself. The interplay between its utilitarian nature, its artistic beauty, and the logic and fun derived from proving results and understanding concepts. I thoroughly enjoyed the intellectual challenge of the mathematics and also enjoyed its applicability.

Question 6: I would expect that the same high quality of instruction that my students received in our high school would continue into the university. That is, I would hope they would be challenged intellectually, nurtured personally, and constantly shown the excitement and applicability of mathematics in their university courses. I hope that they would be encouraged to use technology where appropriate to eliminate the drudgery of mundane calculation and also use the technology to better see relationships and concepts being presented. I would hope they would be exposed to professors who cared about each student's success in the class and who valued the educational component of their professorial lives as much or more than the research component of their lives. I would hope that students would have more than just lectures and doing homework in their college experience in mathematics. I would hope that students would be encouraged to be equal members in the learning process and have the opportunity for discussions, both with their peers and with the teacher.

For prospective mathematics teachers I would hope three things would occur:

1. A very strong academic background, hopefully including many courses beyond what they would actually teach in high school. I think it is much easier to teach a course from a perspective of seeing the importance of what you are teaching for further courses. Too many applicants for mathematics teaching positions have far too weak a mathematics background.

2. These prospective teachers should have ample opportunities to ``student teach'' so that they can begin to see those pedagogical and managerial skills that are needed to be successful in the job.

3. Prospective teachers should have mentoring opportunities with successful high school mathematics teachers in order to see what can be done and done well.