Question 1: Students should understand that mathematics is part of the way we think about things and the way we talk about and explain things. Unfortunately, many students come out of high school thinking that mathematics is simply about numbers, letters that stand for numbers, and formulas that tell them how to manipulate numbers to get more numbers. This purely quantitative and arithmetical aspect of mathematics is important--in fact essential--but it must be understood in the context of how mathematics is used by science to model the world as we observe it. It has become fashionable to present formulas from science texts or papers, without any indication of where these formulas come from, or the assumptions under which they apply, in order to show the "usefulness" of math. Students are often asked to plug values of the variables into these formulas to get past, current or future values of some dependent quantity. A ubiquitious example is the collection of records in some track event, to which a line must be fitted and a prediction made. There is rarely a discussion of, say, physiology, or the logic of fitting a straight line to this data. Mathematics education should go hand in hand with science education. The careful analysis and quantification of a scientific problem, the creation of hypotheses, and the testing of the same, are methods of science that have their correlatives in mathematics: examination of cases, algebraization (possibly with geometric visualization), and the application of mathematical reasoning, logic and manipulation. In addition, students must understand that there is another side of mathematics: proof. Generally speaking, hypotheses in science can only be refuted, by new experimental data or better insight, while conjectures in mathematics may, in addition, be proved, i.e. shown to follow ineluctably from the axioms and definitions. This certainty, albeit conditional on the axioms, is the heart and soul of mathematics, and should be part of every student's understanding of what mathematics is about. Furthermore, a proof is not just an exercise in two column formality. A proof is our way of understanding why a mathematical statement is true. Euclid's proof of the infinitude of primes shows us how we can always produce a new prime, even when we think we have listed them all.

Question 2: Students have different personalities, needs and abilities. Very rarely do high school students know "what they want to be when they grow up." Thus, high schools must challenge each student to learn as much in each subject, including mathematics, as is possible. In mathematics, this may entail some sort of tracking, but of a positive kind. Every high school diploma should require mathematics through geometry and algebra; however, time needed to attain this minimum will vary from student to student. Those who can learn algebra in the 7th grade should be allowed to do so; those who must proceed more slowly should also be allowed to do so, but not be forced into terminal "business" or "household" math courses. The minimal algebra course should include the following.

A1. Basic manipulations, such as the addition of simple algebraic fractions, the laws of exponents, and how to work with formulas.

A2. The solution of small systems (1 and 2 variables) of equations (simple equations by hand, more complicated ones via graphing calculator).

A3. Basic principles of logical thinking: conditional statements, quantified statements, negations of these, and the role of converses and contrapositives (without formal logic or truth tables).

A4. Ability to read reasonable "word" problems, identify what is given and what must be found, and write and solve the resulting equations.

For students who are particularly interested in math, science or engineering, some additional topics should be added.

B1. R is a root of a polynomial P if and only if x - R is a factor of P.

B2. Squaring and cubing binomials, the Quadratic Formula and the sum of a geometric series.

B3. The slope of a line and various algebraic, geometric and physical interpretations, including rate of change.

B4. Right triangle and unit circle trigonometry, the Law of Cosines and the formula for sin(A + B).

B5. Exponential functions, logarithms, and examples from the sciences (knowing about e would also be nice).

Above all, students heading into math or science should be prepared to use mathematics as a first resort in attacking any scientific or quantitative problem. This includes the construction of proofs of varying degrees of difficulty, in both geometry and algebra.

Question 3: We'll have many fewer students having to take remedial math or severely watered-down calculus courses. (We won't have calculus students who think that the square root of a^2 + b^2 is a + b.) Enrollments in science, math and engineering will increase. Students will come to us already knowing that math is useful and that they can learn it by working hard, possibly even forming study groups. They won't groan when the word "pr**f" is used, or look at us blankly when we mention similar triangles or the point-slope formula for a line. When we ask them to multiply (2x+3)(4x-2) they won't take out their calculators. They'll be able to read a simple problem, determine what is given and what must be found, perhaps draw a diagram, choose variables using reasonable names, and set up algebraic relationships based on what's given, using appropriate geometry, trigonometry, physics formulas and algebra. They'll be able to solve simple equations by hand and more complicated ones using appropriate technology, and know when their "answers" are ridiculously off. Students will think of mathematics "across the curriculum," seeing patterns and numerical relationships in all of their courses, and applying sound logical thinking everywhere. Every school will have a "Math Club" in addition to, or even instead of, a "Math Team." Teachers, students and outside speakers will give interesting presentations about mathematics: past, contemporary and future.

Question 4: The default attitude of the good teacher should be: "Any student can learn a reasonable amount of mathematics, and like it because it is beautiful, elegant and useful." The good teacher appreciates proofs--though not necessarily formal ones--because they are the soul of mathematics and because they explain why things are true. The good teacher also appreciates heuristic arguments, examples, and some good old-fashioned computation (at least occasionally). Above all, teachers should have a genuine interest in mathematics, extending far beyond the actual topics they cover in class. They should understand everything they teach, but know much more. They should know some science, especially physics, and perhaps some economics. Teachers at any level should be prepared to encounter questions they can't answer and students who are quicker and smarter than they are; in these situations they must work with the student in a positive, supportive manner, pointing the way toward further knowledge. "Gladly would (s)he learn, and gladly teach."

Question 5: In the fourth grade my father explained to me what pi and square-roots are. He read Gamow's "Mr. Thompkins in Wonderland" with me, and later gave me the same author's "One, Two, Three, Infinity." When I was in the 9th grade he bought me Hogben's "Mathematics for the Million," and encouraged me to do all the exercises--which I did, but it took me all year. These were things I could do on my own.