Question 1: Analyzing three phrases helps in considering this question. (a) "High school graduates". Though the majority of high school graduates today go on to two-year or four-year colleges, about 10% get their high school degrees later than their age-group through GED classes or night school, 20% graduate with their classes but do not go on to college, and some who go on to college are not at all interested in majoring in subjects that require substantial mathematics. One cannot and should not expect all these students to take or want to take the same amount of mathematics. We would like to see all students to take through Advanced Algebra, for all college-bound students to take through Functions, Statistics, and Trigonometry, and for all students planning majors that require calculus to study Precalculus and Discrete Mathematics (or the equivalents of these courses).

(b) "Mathematics". School mathematics needs to be broad-based, encompassing all of the mathematical sciences (e.g., statistics, introductions to the mathematics needed for computer science or operations research) and including the consumer and everyday mathematics (e.g., compound interest, annuities, probabilities associated with lotteries).

(c) "Understand". Understanding encompasses at least four dimensions: skill-algorithm understanding, ranging from memorized facts and the application of an algorithm through the selection and comparison of algorithms to the invention of new algorithms; properties-mathematical underpinnings understanding, ranging from the justification of a conclusion using a definition through the derivation of known properties to the proof of a new theorem; use-application understanding, ranging from the application of formulas in the real world through the use of mathematical models through the invention of new models; representation-metaphor understanding, ranging from the drawing of a graph or other representation through the analysis of representations to the invention of new representations. A possible fifth dimension is cultural-historical-social understanding, ranging from knowing who developed some particular mathematics through a knowledge of the causes and motivations for these developments. High school graduates should have knowledge in all these dimensions.

Question 2: Any answer must include more than high school, because what is done before grade 9 has great impact on what happens in grades 9 through 12. It also must be general, because every school has its own mix of students, teachers, and community, and what works in one place may not work in another.

Every high school mathematics curriculum should prepare students not only for calculus but for college-level statistics and discrete mathematics, which may be taken by students before or along with calculus. For this, there needs to be a balance of work with algebra, geometry, functions, trigonometry, probability, statistics, discrete mathematics, and introductory analysis. The curriculum should involve ample experiences with all the dimensions of understanding mathematics and all levels of problem-solving. That is, it should have a balance of work studying algorithms, mathematical theory, applications, and representations, it should give students exposure to large numbers of routine exercises and non-routine problems, and it should balance questions with short answers with those with longer answers, and some opportunity to work on projects that take more than an evening or two to complete. It should ensure that all students are familiar with the latest in calculator and computer technology for doing mathematics and exploring mathematical ideas.

Fifty years ago even without teaching any applied mathematics and with only a smattering of mathematical theory, the first course in college was typically analytic geometry, not calculus. The evidence is that, except for the very best and most interested students, this cannot be done if the study of algebra begins in 9th grade because we expect much more of students today than we did then. Algebraic ideas should begin in the early elementary grades, and a concentrated study of algebra should begin at 7th grade for better students and at 8th grade for average students.

Even courses for the best high school students should not be designed to weed out students, nor should they be designed with mathematics contests in mind. For students who are interested in competitions, there should be a mathematics club or team that meets after school. To avoid early tracking, it is important that a course for younger students not be designed in such a way that an older student taking that course cannot catch up through summer work. A student who does not get into algebra as a 7th or 8th grader should be able with additional work to join up with those students later.

The good high school curriculum has to take the broad range of all its students into account and does not put any of them into dead-end sequences.

For students who need help, there should be rooms set aside before and/or after school where better students are available to help those in need. These kinds of experiences not only help the better students become more proficient, but they also tend to engender a desire to teach mathematics among many students.

Question 3: This is a difficult question with no simple answer. One criterion for success with an individual student coming out of school is that the student would want to take more mathematics even if it is not required by the field of study. Without wanting to take more mathematics, virtually everything is lost.

A related criterion is that the student have a realistic view of what mathematics is needed for certain areas of study. For instance, a student who thinks that the study of business requires little mathematics beyond introductory algebra has been poorly educated. As another example, a student who does not realize that political scientists should know a great deal about statistics and sampling is similarly deficient.

The criterion many people would put first is that the student be competent in the mathematics needed for the student's field of study. This seems like an obvious criterion, but what is needed is not universally agreed-upon, particularly in today's technology-rich environment. Is a student who uses a symbol manipulator such as Derive or Mathematica to factor polynomials worse off or better off than a student who relies on paper-and-pencil algorithms? Is a student who uses a geometry drawing program to explore the concurrency of bisectors of interior or exterior angles of a triangle spending time as productively as one who is proving that the bisector of the vertex angle in an isosceles triangle is also an altitude? One of the distinguishing features between high school and college mathematics is that there is almost always a time crunch, and the priorities may be more determined by philosophy than able to be determined by research studies.

Part of the difficulty in answering the original question is that, from a successful program, at the next level one tends to have students who were not as good or as interested as when the program was unsuccessful. Perhaps an analogy will better explain this statement, which may seem to be counter-intuitive. Compare a high school with a thriving orchestral program, with large numbers of students taking music lessons on the outside, with a second school that has no such program. From the thriving program comes many students interested in continuing their orchestral experience even if they are not music majors. From the second school the only orchestral students who come are those whose interest was so great that they endured despite the absence of a good school program. The average quality of students from the second school might be higher because there are so few.

Transferring this analogy to mathematics, from some high schools most of the mathematics students we see at the college level are the survivors of a poor program, those who persevered despite poor teaching and a narrow curriculum, or those who needed to go outside their school to special programs in order to find worthwhile mathematics. From other high schools what we may see are students who love mathematics, who were successful at it, and who may even want to take college mathematics for fun. These students will not take well to an environment in which the goal is to weed out future mathematicians from those whose interests are more dilettantish.

Question 4: At Chicago we only train prospective mathematics teachers at the master's degree level and we require a bachelor's degree or its equivalent. A broad mathematical background is essential: some analysis beyond multivariable calculus; linear and abstract algebra; some advanced Euclidean and non-Euclidean geometry (a common deficiency even in students coming from the best schools); number theory (often a deficiency); foundations (logic and set theory); history of mathematics (another common deficiency); both theoretical and practical statistics; probability; familiarity with an advanced programming language (C++ is preferred today); some discrete mathematics such as dynamical systems. We also recommend topology, applied mathematics to virtually any field, and numerical analysis. No student comes in with all of this, and so in our teacher-training program we require six courses in the mathematical sciences and try to fill in gaps in the background.

Teachers must have a positive attitude towards their subject and towards what students can learn. They must also have a broad view of the subject; for example, teachers who think that geometry is worthless or that algebra is merely a bunch of rules or that everything is functions or that calculators should not be allowed in classrooms hurt students by distorting their education and narrowing their vistas. Teachers must also be tolerant of the differences among students that will affect student interest, student performance, and the amount that a student can work in the class.

We do not know what pedagogy or pedagogies work best for a given student, but we do know that students learn most when they are actively involved in their learning, when they are doing mathematics and talking about it rather than passively listening. Like adults, students need variety, so lecturing all the time, or discussing all the time, or putting students in groups all the time will not be as effective as variety. Furthermore, some mathematical topics lend themselves to lecture (e.g., definitions and summaries), others to group work (e.g., analyses of data), still others to discussion (e.g., alternate proofs), so it is not wise to think of only one approach as being paramount.

Question 5: Finally a question to which I can give an answer to which no one else can disagree! In 8th grade I decided that I wanted to be a teacher. In 9th grade I had a mathematics teacher who was the scourge of the school. Virtually everyone was afraid of him, and because he taught most of the advanced mathematics courses, he caused even some of the best students in the school to avoid mathematics. When I received an A from this teacher, I figured I must be good at mathematics and so I thought I would teach mathematics and try to do it in such a way so that students would not be turned away. I was a good student in virtually all subjects, but later in high school I won a number of awards for mathematics and this convinced me that I could carry out my 9th grade decision.

The other factor that caused me to want to major in mathematics was my love for the subject. As I would learn theorems one after another, I would be amazed by their beauty and by the power of mathematics. I recall that as a high school junior, I wrote an essay for an English class on the beauty of $e^{i \pi} = -1$.

In college, I was quite successful in mathematics; I competed on the Putnam team, and my bachelor's thesis was published (in the Annals of Mathematical Statistics). Because of this, there was no doubt either in my mind or in the minds of my professors that I could go on for a doctorate in mathematics, and they tried to talk me out of going into mathematics education, but I felt that I might be able to make a significant contribution to the teaching of mathematics and though I knew I could derive and publish original mathematics, I felt that the contributions would probably not be particularly significant.

I should note that I went through my undergraduate work without a knowledge of any applications of the mathematics I was taking other than to physics and to probability situations. It was only after I was on the faculty at Chicago that I came to realize that the field of applied mathematics was at least as large as that of pure mathematics, that mathematics would not have its importance in the world were it not for its applications, that without its uses it would be treated like chess (and how many universities have chess departments?). Henry Pollak and others convinced me that the boundaries between pure and applied mathematics are not sharp, that there is beautiful and elegant mathematics in both; messy and inelegant mathematics in both, too; that one is not necessarily more or less difficult than the other, and so on.

Question 6: (a) Students take mathematics courses in college for three reasons. Either (1) they want to major in one of the mathematical sciences, (2) they like mathematics enough to take it as an elective, or (3) they need to take mathematics for some other major or for graduation.

We who are in higher education have a duty to light fires and keep fires kindled for those who are in category (1), to increase the number of students who are in category (3), and to ensure that it is not because of us that students in category (2) decide not to major in some area that needs mathematics. We need to be ecumenical in the sense that we need to recognize that there are areas other than mathematics that have need for students very well-trained in mathematics and not to view our students as lost should they decide to switch into those areas; we should delight in this explosion of mathematical language and applications. We also need to be ecumenical in the sense that we do not view it as beneath us to explain our subject to those who are not majoring in it and even to those who are not very good at it. If we do not explain our subject to others, who will?

(b) For prospective mathematics teachers, we need to model good teaching. Good teaching includes clear explanations and questioning and discussing in class; it is two-way between the teacher and the students, not one-way from teacher to students. A good instructor teaches the subject and about the subject, providing motivation and history and applications of the subject and never saying "You'll learn why you are taking this course in the next one." The good teacher is fair, letting students know what is expected of them and recognizing that the students in the classroom have different backgrounds and may need different approaches and amounts of work in order to succeed. Finally, the good teacher is supportive, encouraging the student to succeed rather than setting up situations for failure.