Question 1: Most of all, I want students to appreciate the empirical nature of mathematics. They should know that when we do mathematics we are interacting with an important part of the real world. Like the rest of our world, mathematics can be explored and discovered. We can be creative with it and enjoy its beauty, we can do experiments with it, we can tinker with it, and we can engineer it and use it to create useful technology to improve our lives.

Once students understand that mathematics is part of our experience of the real world, they are more likely to explore things for themselves, to make common-sense checks of their assertions, to develop their own ideas and methods, and to share their experiences with other students. They are also more likley to believe that they can confirm their ideas with reality checks independent of the authority of the book or the teacher. They are less likely to be trapped by the view of mathematics as just a language--that if one can only push the symbols around the "right" way, then the "right" answer will come out in the end.

Question 2: Exploration is at the heart of mathematics. I see great value in the idea of identifying a spectrum of significant open-ended problems that will engage students in the process of doing mathematics for themselves. This should not be done as a replacement for teaching the basic skills students need for their daily lives, but rather as an enhancement. When students learn mathematics through exploration, they experience a side of our subject that lies deeper than the simple skills of algebraic manipulation and calculation. They come to better appreciate the human side of mathematics and to feel that they can invest some of their own personality in the subject. If we emphasize the process of discovering mathematics along with the results of those discoveries more of our students may also develop mathematical tastes favoring hard problems.

I would like to see more emphasis on the use of good mathematical language. College students often do not understand the standard mathematical language that we use in the classroom. A stronger high school preparation in logic and reasoning might be helpful.

In general, high school students should learn good habits of mathematical thinking--experimenting with examples, asking questions, looking for patterns, formulating conjectures, using language accurately, and numerous other standard techniques of mathematical exploration. Long-term projects might provide students with good opportunities to practice some of these techniques.

Question 3: This is a hard question. It is much easier to recognize when individual students have obtained a good high school mathematics education. Such students are robust problem solvers, they are confident of their ability to use mathematics in non-mathematical settings, and they are able to communicate their ideas accurately. They are curious about why things work and when they don't work they are able to analyze problems for themselves.

Evaluating the general quality of mathematics education seems to be much more problematic. Standardized tests have their limitations, but when they are properly designed I think they do tell us something about the quality of education our students are obtaining. It has been argued that standardized tests encourage teachers to teach to the test with the result that our students are being rigidly trained for the narrow experience of test-taking. This is plausible and certainly deserves attention. However, if it is also true that student scores on standardized tests are declining, then perhaps standardized test results are accurately telling us that teaching rigidly to the test is ineffective.

Question 4: In order for teachers to share the experience of mathematical exploration with their students, it is necessary that they have had significant experiences of their own. To successfully foster good habits of mathematical thinking in their students, teachers need to have first internalized those habits in themselves. Our perspective, as research mathematicians, on the nature of mathematics is perhaps the most valuable contribution that we can offer the teaching profession. Our experience working with teachers in Boston University's PROMYS program confirms that teachers are eager and anxious to learn significant mathematics in innovative ways, if only we are willing to help.

Question 5: I have always been interested in all of the sciences, but what attracted me to concentrate on mathematics was the fact that it could be done without a laboratory or special equipment. This made the subject accessible to me as a youngster in ways that the other sciences were not. I found that I could actually do mathematics myself, but in the other sciences I could only read about what others had done.