Question 1: I care most about their understanding of mathematics as a process for analyzing and solving problems of consequence in the natural and social sciences. In particular I want students not to approach mathematics as a "content-free" exercise in manipulating symbols via memorized rules. I will gladly trade some manipulative skill to have students who approach mathematics as a discipline that requires understanding of the concepts and creative thinking to apply those concepts in the solution of problems that matter. Students should not freeze when confronted by a question that goes beyond the "template problems" of their textbooks. They should view mathematics as a toolbox and be able to select intelligently among their tools. (This is closely related to the "Rule of Three," the analysis of problems graphically, numerically, and symbolically.)

There are, of course, basic manipulative skills that high school graduates should have. Most mathematicians would probably put together pretty much the same list and in the same order of priority; the primary differences would be in the lengths. My list would be among the shorter but with the requirement that students understand these topics well. When it comes to skills, real ability with a small collection of tools is far better than superficial acquaintance with a large collection.

Question 2: The essential feature of every high school mathematics curriculum should be the existence of a mathematical environment which instills the attitudes cited above. Courses should not emphasize the memorization of manipulative rules and their regurgitation on template-problem exams. Students must realize that mathematics is thinking, not mental gymnastics. Time should be taken for real applications. Better yet, real applications should be used to motivate the need for certain mathematical techniques. For example, trigonometric functions should be introduced not solely to "manipulate stuff with triangles" but as the primary tool for dealing with repetitive, periodic phenomena. Analytic geometry should be stressed, not simply as a collection of skills to master, but as a profound link between algebraic problems and geometric problems. (Perhaps I missed it, but I don't recall these viewpoints in my high school education. My current students certainly do not come equipped with them.)

I think that high schools need to be very careful about instruction in calculus--I do not think it should always be the capstone goal for their best students. Getting to calculus in high school often means rushing students through the foundational material and developing a "template-problem" mentality to all of mathematics due to shaky understanding of the basics.

Emphasis on varied and effective pedagogical techniques needs to be given as much attention as the curricular content. It is my understanding that great strides have been made in this area in recent years. "Chalk-and-talk" is no longer the only teaching format in the high school arsenal. That's absolutely critical. "Telling is not teaching; hearing is not learning" is a phrase to live by. Collaborative learning situations, both in-class and out-of-class projects emphasizing multi-step problems, discovery learning through calculator or computer labs, written reports--all of these instructional methods should be part of the high school experience. The shift must be towards more opportunities for active learning rather than passive.

Examinations must put more emphasis on conceptual questions and problems requiring thinking, not just template problem memorization. I know this is hard. I have been wrestling with this problem for years at the college level. But the cliché that "exams define the course" is true and cannot be ignored.

Question 3: We will know that high school mathematics education is working well if students emerge with the attitudes about mathematics described above and with sufficient computational ability and content knowledge to succeed in their next level of scientific or mathematical education. These goals are not easy to measure with a simple tool such as a comprehensive examination. We must constantly ask ourselves if the instructional techniques we use are going to move our students in the directions pointed to by these goals. Our yardsticks need to be the current research findings on how students learn mathematics, the observations of those colleagues who deal with the students at their next level of education or job training, and our own instincts as educational professionals. Unfortunately most of us tend to rely almost exclusively on the last mentioned yardstick. Perhaps the professional societies could increase their activities to help bring the necessary information to high school and college instructors. (The demise of UME Trends was an unfortunate backslide in this effort.)

Question 4: Secondary school mathematics teachers must view mathematics not solely as a collection of facts to be transmitted but as a process for analyzing problems which places a premium on the creative choice and application of tools in novel situations. It is this philosophical overview--along with the attitudes I've described in answer to the earlier questions--that is often lacking with many instructors of mathematics. Nothing but cosmetic change will result in mathematics instruction unless a process orientation replaces fact-transmission as our central paradigm. Both are necessary but the emphasis in our current practice is clearly misplaced.

I would like all secondary school mathematics teachers to have been mathematics majors--or at least minors--in college. The particular mathematical content of their courses is not crucial (although exposure to abstract algebra, multivariable calculus, and probability and statistics is ideal). Of more importance is that pre-service teachers be taught via methods that we wish them to emulate in their own classrooms. In this regard we college instructors are much to blame for the overuse of lecturing in secondary school classrooms--it is far overused in college classrooms. The education courses that pre-service teachers take should emphasize the use of a variety of instructional techniques and should present the latest research results on how students learn mathematics.

Question 5: I liked playing with mathematics--doing it--and was good enough that I received a lot of encouragement and recognition in high school. Although my high school experience was with a standard curriculum using standard teaching techniques (i.e., lectures), I had teachers who cared about mathematics, were knowledgeable, and showed a real excitement about the subject. Although my instructional preferences might belie it, I am not an applied mathematician. My area is Lie theory, a topic that I love because of its inherent beauty and its ties to so many branches of mathematics.

A criticism often leveled against the instructional reorientation I recommend is that it will not appeal to those students who, "like us," will become interested in mathematics for its own sake and who would languish under a "less rigorous" curriculum. Not so. I wish my own secondary education had been oriented more to conceptual understanding and applications; I think I would have achieved a healthier and more comprehensive foundational view of mathematics and would have ultimately developed as a mathematician much faster. My experience with teaching beginning level mathematics majors confirms these beliefs. Take the time to set the right foundational attitudes and all our students benefit.