Question 1: I want students to understand that mathematics is a live and dynamic subject which is growing by leaps and bounds through research and new discoveries on a daily basis. They know that mathematics is accessible to many students who have not been included in the past, including minorities and women. Mathematics is a process of ongoing discovery and verification of results which is based on agreed-upon assumptions. Discoveries of great significance have been made in the past, and students know about the historical context of mathematics. Students know how to start on a problem which is completely new to them. Students understand the value of playing with examples, looking for patterns, and trying to generalize. There are a variety of tools which should be known by a student for him/her to be most effective in learning existing mathematics and discovering new mathematics. I want students to know basic algebra, geometry, trigonometry and calculus, but to know these subjects from an application orientation. They would know more than algorithms and theorems. They would understand that mathematics can be used to model many real-world phenomena. They would be able to access appropriate knowledge to solve problems and create new models and new problems. They are able to use resources to look up facts and other items which are not in their memory bank. They understand the appropriate use of technology as a tool to generate new understandings and communicate findings. They are persistent and understand the effort required to understand many aspects of mathematics. Students should have developed confidence in their own ability to do mathematics.

Question 2: A high school curriculum should systematically expose students to concepts of number, algebra, geometry, data handling, uncertainty, trigonometry, change processes, dynamical systems, calculus, etc. An ideal curriculum would blend these subject areas and use concepts from any of them as necessary to solve the problems posed. The curriculum should be based on significant problems which are of interest to students. In the process of understanding these significant problems, they should develop skills in mathematical processes which need to be retained. The curriculum should include opportunities to learn and use technology in the understanding of mathematics.

Question 3: Perhaps the best answer is, when people quit asking the question. I do not believe that standardized test results are an adequate measure. I do not believe that the impressions of college mathematics professors are an adequate measure. High school mathematics will be considered successful when all partners in the education process collaborate to assure that students are prepared for a technical world fraught with unknown circumstances and unplanned events. This requires a commitment from society at large to support education in a variety of ways. Family support is the initial condition. Adequate resources to support a quality education is next. Job opportunities for successful students must be available. The work of mathematics and mathematics-related careers must be understood and valued by society. The role of mathematics in being a productive citizen and understanding things like voting and budgeting must be advanced. When these factors are present, students will have motivation to be successful mathematics students. In effect, high school mathematics will be as successful as the society which surrounds the high school.

Question 4: I believe that the mathematics background for a high school teacher should be a full mathematics major with a broad base of courses. This is essential for a teacher to be able to pose interesting problems and be prepared for any direction which the students may pursue. Teachers must have confidence to permit students to explore, conjecture, and dream about mathematics. Teachers must love their subject. You cannot fake that in a classroom. If you do not have that quality, go into engineering! The most important pedagogical skill for teachers is to pose good problems, ask leading questions, and encourage a variety of solution strategies. Rarely is there a single method for solving a problem. Some of my most exciting days have been when a student creates a process for solving a problem which is simple and more elegant than any I had thought of before. Teachers must understand mathematical modelling, persistence, and verification of results.

Question 5: I loved the challenge of figuring out something that I did not know before. I had the opportunity in my beginning algebra class to work ahead of the class. This meant that I did not have the ``advantage'' of lectures on what I was supposed to do or how I was supposed to do it. I thereby invented many of my own procedures. This was exciting for me. My father had me sit at his elbow every year at tax time and help him with the returns. I learned to use the old mechanical calculator, which now sits on my curio shelf, at an early age. I was fascinated at the chains that were dragged in each column and how the next column was affected when the number got past nine. The farm life lent itself to many problem-solving situations in repairing machinery, etc. This just seemed a lot like math to me, because the procedures to disassemble something usually had an inverse procedure to reassemble. The importance of sequencing was learned early along with geometric properties.

Question 6: a) I want them to be affirmed in what they know, instructed in what they do not know, and valued as colleagues and co-learners in the process of learning. I want them to be nurtured and supported in their learning and not discouraged by evidence of egotism and self-importance. Colleges cannot remake students, so are obligated to accept them where they are and take them to the highest levels possible.

b) Model excellent teaching. Do not pamper them. Require the same rigor and excellence that you require of all students. Talk with them about your goals and methods of teaching. Use hands-on materials to illustrate concepts. Use technology in appropriate ways as a model for your students. Remember, teachers tend to teach as they were taught. College teachers have a tremendous responsibility in this regard.