Question 1: Students should have the basic knowledge and skills of algebra and geometry. This includes the properties of numbers and functions, and good computational skills with them, both with and without calculators, and good judgement about how and when to use the latter. It also includes knowing important general formulas and algorithms, and, in some cases, how to derive and/or analyze them. In geometry, students should know basic euclidean geometry, cartesian and polar coordinates, elementary properties of the trig functions, and some initial graphing techniques.

Students should know how to solve interesting and natural mathematical problems, and enjoy doing so. This involves skills like elementary modeling (as conveyed in good word problems), pattern recognition, data analysis, back-of-the-envelope estimation, solution strategies, supple use and invention of mathematical notation,...

Students should know the various levels of mathematical reasoning, from heuristics and conjectures, based on past knowledge or found examples, all the way to rigorous formal mathematical proof, and to understand the significance and appropriate use of each.

Students should have a good command of oral and written expression in English, understand that mathematics is conveyed with very precise language that has to obey, even more than ordinary speech, the rules of grammar and syntax, and that mathematical expressions, even those with symbols, must be expressible as whole, proper and unambiguous sentences.

Question 2: The answer to this is implicit in the answer to 1. above. I consider the above matters an essential core. A curriculum might well go further in one or more directions--calculus, statistics, discrete math,... I favor a curriculum that develops sound strength in fundamental subjects and techniques over one that escalates up the mathematical hierarchy. The latter often produces knowledge that is shaky in college, and has to be retooled there.

Question 3: We first need to achieve some consensus on the aims of that education. They are multiple--cultural and intellectual (college performance), economic (workplace performance and employability), and social (informed and responsible citizenry). We need some agreement on what balance of priorities these are given, and on whether and how the student population should be aggregated in delivering that education. Once we have some semblance of consensus, including a good and collaborative articulation between high school education and college and the workplace, then we can go beyond the current system of national exams--whose value and relevance are conditioned by the extent to which what they measure conforms with the educational aims of the system--and do longitudinal studies of performance of high school graduates in college and the workplace, not only in terms of course grades, but even of their remaining engaged with scientific and technical subjects.

Question 4: Ideally, high school teachers should have a very solid and broad knowledge and understanding of mathematics, through calculus and ODE, some modern algebra and discrete math, statistics, and some history. They should show confidence and pleasure in communicating the subject to students. They should have a good command of diverse pedagogical techniques, ranging from group work and open-ended project work, perhaps technologically based, to more formal and disciplined traditional exposition. Likewise, diverse forms of student assessment should be part of their repertoire. The individual teacher, so qualified, and in consultation with other teachers, should make decisions about what mix of teaching styles and techniques are appropriate at each moment, and with each student population.

Question 5: I always enjoyed math at school, but had little sense of its scope until my brother, Manuel, in a Navy officers training program in WWII, came home on leaves and gave me enthusiastic tutorials on the science and engineering courses he was taking. He continued this later as a student at Caltech. When I attended Princeton as an undergraduate, the honors calculus course was run by E. Artin, with Lang and Tate among the instructors. The excitement of that environment won many of us over to mathematics.