Question 1: High school math has different goals. For students who will use math later on--math, science and engineeering majors--high school math should prepare them for more advanced stuff. For these students, technique is fairly important. For others, an appreciation of certain basic ways of thinking would be a satisfactory outcome. It is hard to accommodate these differing goals within one program.

Question 2: I actually have some fairly precise ideas about how I would like to see the high school curriculum change, but that is too complicated to get into here. So these choices are based on my understanding of the current curriculum. They are for the science-directed group. Algebraic technique: facility with rational arithmetic, including division and the law of exponents and the Euclidean algorithm; appreciation of the analogy between rational arithmetic and decimal arithmetic, especially for division; completing the square and the quadratic formula; the geometric series; Pascal's triangle, a^2-b^2; systems of linear equations (2x2 and 3x3). Geometry/trig: straight lines, distance, angles; triangle inequality; addition of angles, perpendicularity; angle sum in atriangle; Pythagorean Theorem; geometry of triangles and circles, law of sines and cosines; addition formulas for trig functions, similarity, the basic facts about complex numbers. Beyond this factual/technical base, one would like to have an appreciation of certain aspects of mathematical culture. First, the importance of precision and careful reasoning, with minimal assumptions, that is, the ideas of definition and proof. This would include an appreciation of the field axioms, and their usefulness in rational arithmetic, and a study of the basic facts of syllogistic reasoning (inverse, converse, contrapositive, proof by contradiction). Of course, these days, most of the best students see calculus in high school. I would like to see a lot of attention paid to the idea of limit here, but I don't think it happens much. I am not sure exactly where to say this, but I think it important to emphasize that mathematics should not be conceived of as an isolated subject. It should get used. While some applications can be done in math class, math should not have to carry the burden of this. Math should routinely get used in other subjects. This of course is not speaking to the curriculum as it stands.

Question 3: One way would be for there to be an increased acceptance of mathematics as part of normal dialogue. In the book Mathematics Tomorrow, edited by Lynn Steen, there is an article by Neal Koblitz called "Mathematics as Propaganda." It starts with a story about someone from Zero Population Growth talking about environmental impact as being proportional to population. This was on the Johnny Carson show, I think. The guy wrote a linear equation, D = cP, D being damage and P being population. Koblitz's point was, writing this equation was an intimidating thing to do. My reaction was to be depressed that something so basic and simple should be perceived as intimidating by the broad public. If we could reach a state where this kind of thing was an accepted and understood part of normal discourse, that would be a positive thing. But I guess that won't happen in the short term. I think the practical short term answer is to recognize, first, that "working" will always be a relative term; one can always wish for more, and things could always be worse. Given that, the answer becomes technical: one must have standardized tests which reflect a consensus view of what needs to be known, and measure progress on these tests. There is another criterion, in terms of manpower: are the math skills offered on the market adequate to the jobs available? But this is a much more contingent criterion, reflecting much more than US education. For example, there is currently a math PhD supply glut. But these supernumerary PhDs are not US citizens.

Question 4: I think,first, teachers need to love and understand math as a culture, and the importance of that culture to civilization, and the importance of transmitting it, which is their job. This would include a historical perspective, so that they can convey not just the technical aspects, but why the points of view embodied in the technical points are valuable, and how they represent, as many of them do, triumphs of human thought. Of course, I would want teachers to be well-versed in mathematics itself, to the extent that they are comfortable enough with it to present it as a living thing, a set of responses to challenges, refined over the years. This involves a much deeper understanding than just being able to do the problems. Relatively subtle issues can be expected to arise on a frequent if inividually unpredictable basis, and a good teacher would be ready to discuss and clarify many of these issues. Specific pedagogical techniques can vary with the teacher and the situation, but practice, reflection on practice, and discussion with others of specific points can lead to improvement. Most learning that takes place is due to student effort. The task of the teacher is to bring mathematics to the student in learnable chunks, and to put the student in a frame of mind to master them.

Question 5: I was identified as "good at math" long before I had any idea of a career related to it. A fifth grade teacher told me I would be a mathematician, and my (private) reaction was, "You're crazy, lady." In sixth grade, I rejected a proposal to study math beyond grade level. In tenth grade, several things happened. I read a popular book about quantum mechanics. It had some mathematical symbols (line integrals) in it, and I had no idea what they meant, so I started to study on my own in order to understand them. For a long time after, through much of college, mathematical physics was a big motivation for learning math. Also in tenth grade, I took geometry, which I thought was great. Also, one of my geometry teachers was a real enthusiast, and put me on to the books Elementary Mathematics from an Advanced Standpoint by Felix Klein. They were very hard, and I didn't understand a lot in them, but they showed a completely different world from high school math.