Question 1: Before discussing what mathematics high school students should understand, let me point out that many students graduate from high school believing that understanding mathematics is not important or not possible, at least for them. However, the thing that I most care about is that they do understand, at some level, the mathematics they have learned. At what level they understand may vary from student to student.

What areas of mathematics they know is indeed of concern to me, but I put understanding ahead of knowing particular topics. I know from experience that people who have seen many topics, but who don't really understand them, are much harder to teach than those who have seen fewer topics but who understand them. A person who can explain why he/she thinks something is true, who is resourceful enough to decide whether or not an answer is correct--this is a person who can learn easily. It is the habit of trying to make sense out of what one learns which makes a person educated.

Question 2: The curriculum sketched below is realistic in that it could be taught in any US high school before the turn of the century--and it could be taught to most students. However many students can, and should, go beyond the topics listed into calculus, statistics, linear algebra and differential equations.

Arithmetic, both by hand and using a calculator. Particularly percentages.

Algebra, especially using a variety of symbols (not just x and y). The learning of algebra is going to be affected by the fact that calculators can now do symbolic manipulation. Since we still want students to understand algebra, there are two possibilities. We can either leave our curriculum the same and ban calculators or we can remake the courses. If we don't remake the courses, students will do it for us. They will use calculators outside of class even if they are not allowed to use them inside. As a result, their traditional paper-and-pencil skills will decline and they will not have acquired anything else from our courses. Since we do not want students to do everything on a calculator, remaking the courses in such a way that, even with a calculator, students learn algebra is the main challenge facing high school mathematics education today.

Graphing, including data presentation. The ability to read and interpret graphs as well as draw them.

The geometry and trigonometry related to measurement--lengths, areas, volumes--as well as geometrical reasoning. The use of logarithms to understand compound interest, present value, and exponential growth.

The basic ideas of statistics. Writing an argument using graphical or numerical data as evidence. Numerical literacy in assessing news about social and economic problems. For example, being able to evaluate statistics and graphs on unemployment, inflation, and environmental risks.

Throughout the curriculum there should be problems in which the student is asked to apply or interpret the mathematics in ways that he/she has not seen before. How difficult these problems are will depend on the setting, but some are always appropriate.

Throughout the curriculum students should be expected to write convincing explanations. This will ensure that all students take at least the first step along the road from clear reasoning to precision to rigor.

Students should take mathematics every year so that they don't arrive at college having forgotten what they learned.

Question 3: College faculty in the classroom will know high school education is working well when students exhibit the kind of understanding described in #1.

Exams and test scores tell us something too, though they are sometimes unreliable because teachers can teach "to the test." A more reliable measure is a test given a long time after a course. A student who understands material nearly always remembers it for a period of time.

Successful high school programs will produce some payoff in student skills, but more in attitude: persistence, independence, the ability to check one's own work, the willingness to tackle new problems. Lack of computational skill in a high school graduate can be a problem, but strong computational skill does not guarantee success. Some computational skill is necessary; lots of it is in no way sufficient.

My main problem in answering this question is that, at the moment, college faculty are often unaware of what their students understand about mathematics. For example, many faculty give lectures without realizing that a large number of their students do not follow. Faculty who are attuned to students' thinking processes are better equipped to recognize when education is working well.

Question 4: To achieve the kind of understanding required in #1, teachers must have it themselves. Thus, in addition to having gone beyond where their students must go mathematically, teachers need a crystal clear picture of how the mathematics they know makes sense. Their understanding needs to be strong enough that they can recognize any beginnings of understanding in their students, no matter how original or how poorly described. Teachers need to be confident enough to argue about mathematics with their students and their colleagues. They must know how mathematics is used if they are to convince students that learning mathematics is useful.

Teachers need not only to understand mathematics but also to understand students: how to reach them, how to challenge them, how to affect their thinking processes. Teachers need to have the charisma and self-confidence to earn the respect of the students and the intellectual strength to be able to exploit the fact that their students are listening.

Question 5: Finding the same mathematical structure in different contexts, both inside and outside mathematics. For example, ideas that seemed remarkable when I first met them: the relationship between the solution space for a differential equation and vectors in a plane; between complex exponentials and the behavior of oscillations; between sums of trigonometric functions and beats; a geometric interpretation of the concepts of linear algebra; point set topology and limits; Maxwell's equations and the relationship between electric and magnetic fields.