Question 1: I care most that high school graduates understand that mathematics is accessible. I want students to know that they can "do" mathematics. Currently, such self confidence is shared by a painfully small minority of our high school population. Large numbers of students dutifully go through the paces, earning quite acceptable grades. Yet, when they are confronted with a problem that bears little resemblance to problems they have seen before, there is a nearly universal conviction that the solution is a mystery that cannot be cracked by ordinary mortals such as themselves. I recall my first experience with a federal income tax form in the mid-sixties, when they were more complicated than they are now. Daunted, I remember reassuring myself with the mantra, "I have a master's degree in mathematics. I can do this." I want my students to have such confidence in their ability to respond to the day to day requirements of this technological world without having to first major in mathematics. I want them to feel free to choose college courses whether or not such courses have a heavily quantitative component. I want them to apply for jobs knowing that they have problem solving skills, that they can respond to the challenges of a new position.

Question 2: In the introduction to the National Council of Teachers of Mathematics Standards for School Mathematics (1989, Pg.4)), Henry Pollak is quoted as listing the following mathematical expectations for new employees in industry: " The ability to set up problems with the appropriate operations; Knowledge of a variety of techniques to approach and work on problems; Understanding of the underlying mathematical features of a problem; The ability to work with others on problems; The ability to see the applicability of mathematical ideas to common and complex problems; Preparation for open problem situations, since most real problems are not well formulated; Belief in the utility and value of mathematics."

If we could teach our courses so that our students could meet these expectations, they would be prepared for life in the 21st century as well as a future in industry. They would also be ready for the demands of advanced studies -- whether in mathematics or in its client disciplines. To prepare students to meet these expectations, every high school mathematics curriculum should:

a) include open-ended questions in exercises;

b) require that students employ estimation and common sense to get an idea of a reasonable result before they plunge ahead with familiar algorithms;

c) incorporate activities that require cooperation;

d) give theory a context by providing meaningful applications, visual representations and manipulative opportunities (including use of technology);

e) incorporate long term projects that i) give students a chance to bring their creativity and own interests into the mathematics classroom or ii) involve exploration of an idea, or iii) are problems that require higher level thinking, time and cooperation to resolve and which can be solved by a variety of approaches;

and f) emphasize that explaining the process by which one arrives at a solution is as important as the solution itself.

Question 3: Teachers would know when mathematics education is working well when they

a) can give a problem which does not resemble any that their students have seen before and not be met with blank stares or protests of unreasonableness;

b) receive papers in which students are able to write about concepts and explain processes with comprehension and lucidity;

c) see students taking mathematics courses because they want to, not because they have to;

d) get "why" and "what if" questions rather than "just tell me what to do" statements; and

e) have as many girls, young people with handicaps and students from minority groups in higher level mathematics classes as in the lower level ones.

To some, this might seem like an unrealistic wish list. Yet I have seen first second and third graders be excited about mathematics, be able to come up with solutions to interesting -- and, often, open-ended -- problems and proudly explain their reasoning. In twenty years, we will know if mathematics education has been working when we can go to a cocktail party, tell a stranger that we are mathematicians, and not get a response of veiled distaste ("That was my worst subject!") or muted awe ("Are you really?"). While I certainly enjoy the respect granted me as a practitioner of the mysterious, we can no longer afford to have only a small minority of this country privy to the workings as well as the beauty of mathematics.

Question 4: I am preparing to co-teach a graduate level methods course for aspiring middle and high school mathematics teachers. Reading my class list, I find myself hoping for students who really enjoy mathematics and who like young people. If one has a passion for the subject, and enjoys sharing that excitement with teenagers, the combination is hard to beat. As I consider the mathematical backgrounds of these incipient teachers, I have short term and long term concerns. For the immediate future, I think all high school teachers need to have a solid grounding in and an appreciation of calculus, since most will be teaching a curriculum whose current capstone is that course. While it would be nice if all teachers had backgrounds in statistics (material that all high school students should be exposed to), discrete mathematics and the history of mathematics (to provide their students with context), I am less concerned with specifics beyond calculus than I am with a grasp of the mathematical process and a facility with the language of symbols. With such literacy, one can learn whatever new material one might require for a given course.

As I consider the mathematical background of people who are entering teaching now, I am keenly aware that we cannot imagine what the world is going to be like for our students thirty years from now. If, indeed, as much mathematics has been created in the past 50 years as in the the previous 3,000, traditional topics as well as ways of teaching are going to become irrelevant. We need to be coaches, helping students to expand their problem solving skills, not sources of great knowledge whose mission is to fill empty heads.

Question 5: In the early sixties, I attended Wells College, a small, liberal arts school for women. At the end of my sophomore year, I considered majoring in either chemistry, English, or mathematics. I always spent an unreasonably long time completing chemistry labs and so decided that, if I wanted to have any life outside of school, chemistry was not a good choice for me. I also observed a very large number of my classmates majoring in English. Never one to follow the crowd, I turned to mathematics by default. It was a subject that I knew I could do well in, I had enjoyed tutoring other students, and there was no good reason not to.

In retrospect, I appreciate that, had I gone to a co-educational school, I would not even have considered mathematics or chemistry as majors. The perceived masculinity of the subjects -- with the implied competition that I associate with boys and the abstract study devoid of human context -- would have ensured my immersing myself in English literature. At Wells, I was truly free to choose. There was minimal competition among my peers, but there was support for individual interests. I created an independent study in which I was able to explore the applications of mathematics throughout its literature -- a study that made me realize that there was much more to mathematics than I had ever known and that I liked the subject very much indeed.

There is one memory from high school that contributed to my choosing mathematics as a major. My mathematics career in New York City public school was pedestrian, a state that I attribute more to the curriculum and the text books than to the teachers, some of whom were committed and accomplished. I was good at memorization and so got very high grades -- including earning 100% on the New York State Regents exam in geometry even though I had no understanding of what a proof was. But I remember one night in my junior year when I was taking algebra II. I was grinding out my math homework and came across something which did not make sense to me. I spent the next half hour trying to figure out what it was about. Although the source of confusion has long been forgotten, I can still feel the enormous sense of satisfaction that accompanied my announcing to my mother, "I figured it out." I knew at the time that she did not appreciate the enormity of my accomplishment, but my triumph was undiminished. I knew I had done something substantial and I felt great. How sad that my only experience of the power and beauty of mathematics should have occurred despite what was happening in the classroom.

My commitment to and passion for the NCTM standards and the reform calculus movement come from such a history. I believe that, had I grown up in the nineties rather than in the fifties, it is less likely that I would have been limited to one exciting mathematical experience in high school; it is more likely that I would have enthusiastically chosen mathematics as my major, not gone into it by default.

Question 6: To the extent that my students feel empowered as problem solvers and take pleasure in doing mathematics, I would hope that they can continue to grow in confidence and facility as they move on to institutions of higher learning. I would hope that they continue to experience:

a) technology as an integral part of their learning;

b) a variety of ways to learn about a new concept;

c) the requirement that they clearly describe the process as well as the results of their investigations;

d) opportunities to work in cooperation with their classmates;

e) an emphasis on the larger picture of the mathematical process rather than a preoccupation with the details of a specific topic; and

f) substantial problems and long term projects which have meaning for them.

Most important, I want my students to feel enlarged, not diminished (as has happened so very many times), by their experience with mathematics courses on the college level. For many of my students -- especially the girls and students of color -- the spark of interest in mathematics is precarious. I would hope that all of my students might experience mathematics as an adventure not an obstacle course. Given how wonderful the field is, it seems a reasonable wish.