Question 1: I would hope that having studied mathematics for twelve years, a student would face a completely unfamiliar problem with equanimity. He (my hypothetical high school graduate is male for purely literary reasons) would have developed a toolkit of mathematical and problem-solving strategies that would enable him to weigh the problem thoughtfully; to walk around the problem, surveying it, getting to understand it, prodding it, looking for possible entry points, trying out simpler versions, comparing it to similar problems solved in the past, asking friends, breaking it down into manageable chunks, rather than tossing it back like a grenade. I hope that our hypothetical graduate would understand that mathematics is not a collection of tricks and formulas, but is a process that has been refined for thousands of years by some of the finest intellects of the age.

So much for a general attitude. More specifically, I would expect him to have mastered a wide range of mathematical techniques and approaches. I would expect him to be able to sketch an accurate diagram if a diagram is appropriate, to have the geometric and analytic tools to look at the situation from a variety of perspectives, to decide whether coordinates or vectors or synthetic methods are the more appropriate, whether it would be better to enlist the help of a computer or calculator to graph the story or to run a simulation. Even more specifically, I would prefer the student to have seen mathematics being applied to a wide range of problems, rather than forging ahead to ever more advanced calculus. I would hope our graduate would have studied discrete methods and applications, algorithms, graph theory, probability, and statistics. He should have seen that mathematics must occasionally say that no solution is possible, such as fair voting systems, fair apportionment. He should know that mathematics may occasionally say I do not know, as in a myriad of open questions like travelling salesmen problems, tessellation problems. He should know that many disciplines other than mathematics use mathematics, and he should have a sense of how mathematics feels to practitioners of these disciplines.

Question 2: I think my answer to Question 1 may have encroached on this question. An important feature of precollege mathematics is that its focus be broad. A student should not think of high school mathematics as a pyramid, the apex of which is calculus (as many parents do). I have seen too many students struggle desperately through a senior year of calculus with mediocre results. A year of interesting applications, where the actual mathematics is little more than arithmetic, can result in a completely changed attitude to the subject. While there are certainly students who can take a year or more of calculus, most seniors would benefit from a year of topics in discrete mathematics and statistics. A precollege mathematics curriculum should give every student a strong grounding in basic algebra (not exotic factoring examples or eclectic trigonometric identities) and a firm understanding of the concept and notation of the function. A student should be able to interpret and construct graphs. The curriculum should give students the opportunity to work collaboratively and demand that the student communicate his work both in written form and verbally. A high school graduate should know when it is appropriate to enlist technological help and how to interpret the information his calculator or computer gives him.

Question 3: Precollege mathematics is working when a student recognizes when it is appropriate to call a mathematician, in the sense that he knows when to call an electrician or a plumber. It is working when we no longer read dreary self-serving reports that U.S. students have a worse mathematical preparation than students in Upper Volta. It is working when mathematical self-confidence is founded on competence and not on stickers and rewards from the self-esteem industry.

Question 4: If a mathematics teacher has the choice between taking a mathematics course and an education course, there should be no choice. A teacher cannot have too strong a mathematics background. Our ideal mathematics teacher is excited about mathematics and communicates that excitement to his students. His school encourages him to attend professional conferences and workshops; he subscribes to (and reads) professional journals. He does mathematics. In other words, mathematics is more to him than a source of income; it is a source of intellectual pleasure and sustenance. He has a variety of pedagogical approaches. My experience is that students learn when they are engaged and they are engaged when they have to explain and present their work to the class. They are engaged when problems are interesting and not contrived. They are engaged when they are active and not being talked at. They are engaged when they are given the responsibility for their own learning and when they have the sense that their teacher is taking them seriously. An effective teacher will ask students for their opinions. He will make it clear how he is assessing his students, and his methods of assessment will be fair and consistent. A visitor to his classroom may find it difficult to see who is in charge. The chairs will probably not be arranged in rows and columns facing the board. They will be in small clusters or arranged in a horseshoe or in some way that allows eye contact and conversation among students. There will be conversation, questions, discussion, an overall sense of a shared intellectual adventure. There will be models available, concrete materials to illustrate abstract concepts.

Question 5: I can date precisely when my interest in mathematics started. My high school algebra text showed the nets of the platonic polyhedra and a reference to Cundy and Rollett's Mathematical Models. I became hooked on geometry. As a student in the English educational system I was, for a variety of reasons, forbidden to study mathematics beyond the age of fifteen. So I taught myself all the mathematics I could find, geometry in particular, untrammeled by a school curriculum. When I came to America, I took summer courses and correspondence courses and only became a full-time mathematics student as a graduate student at the University of Washington, where I had the good fortune to become a student of Branko Grunbaum. Since I have explored so much mathematics on my own, I think this has made me willing to take risks with my students and put in their way opportunities to explore some less-travelled paths. It has also given me a lifelong (so far) habit of inventing, finding, and solving mathematical problems when I am far from my classroom.

Question 6: When my students graduate, I hope they will be equipped with the tools to make sense of the mathematics they meet. I expect they will find classrooms with a very different flavor from their high school--much larger, more regimented classes. They will probably find it less easy to have access to their teachers, who may well be graduate students or mathematicians for whom teaching is a lower priority than research. This is an observation and not a judgment. I anticipate that very few of them will become mathematics majors but that many of them will be forced to take some mathematics courses to complete their major. I expect that there will be a wide range in the support systems that will be available to them when they have difficulties.

If they go on to train to become mathematics teachers, I expect that their professors will be aware of the changes in thinking in mathematics education over the last ten years. From A Nation at Risk on there has been nothing less than a revolution in mathematics education, particularly at the precollege level. I expect professors of education to train students to become teachers who have a strong background in mathematics, are confident in their mathematics, are excited by mathematics, and are prepared to take risks in their classrooms to produce a new generation of students who are not afraid of mathematics.

There is another issue that concerns me but which does not seem to be addressed by your questions. I have mentioned several times that I feel that a broad base of mathematics is more important than scurrying to complete calculus. Indeed, many universities would prefer high schools to leave calculus alone and let them do the job. The Tulane conference spawned dozens of university-based calculus initiatives. And yet there are universities whose admissions offices tell our applicants that without calculus on their transcripts they have little chance of being admitted. There is a huge communication gap between mathematics departments and admissions offices. We are in a difficult situation here at Exeter. We are encouraging average students to take courses in modelling, discrete mathematics, and statistics (the carrot for this is the new AP test) rather than calculus, but they tell us that colleges need calculus. What is to be done?