Question 1: Mathematics is an organized way to understand certain kinds of phenomona: numerical, geometric, algebraic, probabilistic etc. Mathematics provides systematic ways to solve certain categories of problems. Mathematics gets codified in theorems--assertions which can be proved (perhaps in many different ways!) and can be used safely ever after. High school students should see mathematics in at least two ways: (1) a body of facts and strategies for using these facts to solve problems; and (2) a body of phenomena and a techncque (logical deduction) for establishing facts aboaut them; and for those with mathematical insight (3) a body of phenomena which are still open to investigation and codification and possibly use.

Question 2: I would be very happy if high school students brought with them to college the basic ideas that mathematics is a subject matter for thinking and that mathematics is a collection of ways to think about that subject matter. I could do without the idea that mathematics is a set of conditioned responses to key words in mysterious problems leading to steroetypical calculations and culminating in an "answer" in a circle or box. To be more concrete, I would like students to understand how mathematical statements follow from one another; how some statements with variables are true for all values of the variable and others only for some values (or even none); how graphs of functions are related to their equations and how to connect the geometric and algebraic interpretatins of inequalities. To be both honest and greedy, I would also like students to be able to write sentences about mathematics in grammatical English.

Question 3: "Mathematical success in high school" probably has different meanings for differend populations at different points in their lives. The college bound population with interests in science and engineering should be ready for calculus, if they have not already taken calculus and taken an AP exam. Other college bound students should have enough mastery of arithmetic, algebra, and geometry that they can use this material in analysis of statistical data and other interesting topics for students in the humanities and the social sciences. Students should not have to do basic algebra and geometry over again in college after studying them in high school. I like the quality control slogan "Do it right the first time!" The students heading to vocational programs should be able to manage the computations and the geometry implicit in their work. Like the college bound population, these people will doubtless have calculators and computors to handle the heavy computations--but they will need to understand enough about mathematics to choose the right operations to enter, to interpret the results properly ,and to recognize unreasonable output in order to check for mechanical or entry errors.

Shorter answer: High school math works well when high school graduates can use the mathematics they need to get on with their lives.

Question 4: High school teachers must of course be absolute paragons of intellectual and human virtue. They must understand not only high school mathematics, but also its applications in college and the workplace. They must know how to explain what they know in several different ways in order to respond to the different mathematical personalities of their students. They must know how to draw students into thinking and communicating (and dare I add enjoying?) mathematics. They must know how to test the facts and the problem solving and the logical analysis in interesting ways--while conforming to the current fads, and not irritating the parents, and not inflating the grades or deflating the egos of their students, and ... oh yes, tolerating the good advice and lofty opinions of their university colleagues.

Slightly more seriously, I am increasingly convinced of the importance of two newly resurgent pieces of pedagogy. First, getting students to work together. This helps them see that mathematics has meaning and that meaning can be communicated. Second, related to the first of course, is getting students to communicate with each other and with the teacher in both spoken and written form. If students communicate only with the teacher, they will forever fall back on the likelihood that the teacher does really know what they mean. By having to commucate with each other they learn to say what they mean--and they appreciate how hard it is to communicate with a boss or customer or co-worker who does not necessarily know what they mean.

Question 5: I liked mathematics bacause it gave me a way to be very secure about my work. If I read a proof or--even better, made one of my own--then I had a result I could trust. No one could come back and demand that I include yet another point of view or changeable emotion in my essay. I also liked the clarity of analysis and exposition which my teachers helped me to learn. The rhythm of defining variables and restating data in "word problems" provided a momentum which so reliably led to a solution.

And besides I was good at it.

A philosopher of science once pointed out that mathematical proof was powerful in two ways: It convinced mathematicians--and it drove everyone else away from the discourse.