Question 1: I have appended the goals sheets which I give to my students for my mathematical modeling and geometry courses. They are somewhat redundant, but the key consideration is that for most investigators and users of mathematics, there is a necessary synergy between understanding and valuing both applied and abstract mathematics. I seek to help my students understand the nature and skills required for pure mathematics discovery and the application of mathematics to real-world questions (in particular, their own). The goals sheets not only list some of what I consider essential for graduates, they also highlight that you must share with high school students the objectives of their education. They cannot commit to their learning if no one tells them what it is for and convinces them of the worthiness of those objectives. Most importantly, though, high school should be about training thoughtful, inquiring citizens. This proposition suggests that the most important areas of math content are statistics, probability, and functions (e.g., to understand the differing consequences of exponentially and linearly growing phenomena). Without the first two, one is ill-equipped to read the newspaper or judge most policy decisions.

Question 2: The only way to become good at solving real problems (problems which are unfamiliar) is to have continual practice. The only way to get good at posing new and interesting problems is to be required to do so. The best way to learn the mechanical skills of mathematics is to use them repeatedly in the service of answering questions which you want to resolve. (Dewey said that you do not learn the basics by the studying the basics, but by engaging in projects that require the basics.) My students are usually engaged in projects or problem sets of one to four weeks in duration. They invariably pose questions far more difficult than I ever would for them. They care about these creations and become far better at basic algebra and geometry in the process than they would if we focused on rote exercises.

Question 3: a) When all of our high school math teachers like math enough to study it independently, to use some of their time at home to read about new math, play with mathematical problems, etc. No one would accept an English teacher who never read novels, but it is the norm for math teachers to have no connection to mathematics outside of the classroom.

b) When we can provide portfolios of our students' work that demonstrate their ability to use math productively.

Question 4: That they continually learn new mathematics, engage in math explorations and projects of an unfamiliar nature, and they teach so that their students want to learn more math (since one cannot possibly teach all the "prerequisites").

Question 5: My father is an accountant who always did math puzzles with me, and I had marvelous high school math teachers who continually encouraged me and even provided math teaching employment for me from the eleventh grade through my first full-time position. I love studying and doing math.

Question 6: That professors stop lecturing. That they stop seeing techniques as the essence of mathematics. That they acknowledge that creativity plays a role in doing mathematics and believe that their students are capable (at some level) of exhibiting that creativity. That they emphasize learning skills of immediate utility at least as much as those that will be preparation for some other course of delayed gratifications.

I believe there are common cores of experience and content that would serve math majors and teachers equally well. As matters currently stand, neither group is comfortable exploring and posing their own questions. Traditionally taught math is too rote and soulless an activity for all concerned.