Question 1: Obviously, I want my students to see the beauty and excitement in the subject or art form they are learning about. I love hearing comments like "that's neat!" etc. On a concrete level I want my students to have a firm grasp of analysis, and be able to "see" equations as pictures of curves. Students should have an appreciation of the connection between the shape of a function (or relation) and the equation that represents it. This they should be able to analyze both with paper and pencil as well as with a graphing calculator. They should not be totally dependent on the latter and should be comfortable with number facts. (Amazingly some aren't.)

Question 2: Algebra I , Algebra II, Analysis, Geometry, including transformational and coordinate geometry, a strand of statistics and probability throughout, and Calculus and/or matrix algebra for those who have successfully have completed the aforementioned subjects. I ALWAYS recommend that my students retake Calculus when they enter college (usually at an accelerated level if they have taken advanced placement calculus). We have three levels of Calculus: BC, AB, and non-advanced placement Calculus for those who will not sit for the exam. In all these cases we teach to the curriculum of the College Board for the exam. This, incidentally will change rather drastically beginning fall '97 to a more applied less rote curriculum (i.e. integration by parts will no longer be a topic---though I will still teach it---and slope fields will now be included). The change is meant to reflect the reforms that many colleges are making in their calculus programs.

Question 3: Truly not until you have some comparative data, i.e. some sort of national exam a la Europe. For example, the teachers in my department feel our "restructured curriculum" with heterogeneous grouping is failing miserably. Our administrators feel we are being negative (rather than realistic). There will be no "proof" that will be accepted better than our SAT and achievement scores, which we feel will go down (this first group of "restructured" students is just about to take them during the next years) Of course comparison is difficult when the college board "recenters" its scores as it just has done this past year. What my colleagues and I have noticed is that our students are less good than they were five years ago and much less good than ten years ago. We measure our feelings by looking at our expectations of what they could or couldn't do on tests and assignments we used to give. In addition there is more pressure on teachers to pass students (pressure from parents to administrators). In the end, although we are not able to assert our professional expertise as to how to teach students, we are being held accountable for the outcomes of methods which some of us do not believe will work well. In our school we want empowerment to teach students in the very best way. We used to have four levels of math; for example, we taught our Algebra I in levels 2 and 3---the same material but taught in very different ways. Now we have all these students together. It becomes impossible to teach to the top (too many will fail), so both groups get the short end if we teach to the middle. Incidentally, in grading these students it still is very clear that they are made up of two distinct ability levels. We are also concerned with "Algebra for all eighth graders." We find many eighth graders fail their first time through and aren't really ready to deal with the abstractions of algebra until they take it or retake it in ninth grade, when they often then end up with an A or a B.

Question 4: I believe that teachers of mathematics should first and foremost be excited about the subject they are teaching. I would want them to be able to communicate to their students the beauty and the power of mathematics so that students can get turned on to the subject. To do this a teacher must both be strong in her/his subject, keep current in the changes afoot, like integrating the strengths of the hand held computer in their field, and integrating real-world applications along with the traditional subject matter. I would want a teacher to see teaching as a life long learning experience---keeping current on new research, and encouraging students to read and write in mathematics, to see mathematics, and to connect numbers and statistics to their mathematics. To be able to do this a teacher must be perceived by society as a strong and important member in a team, and not a scapegoat or low-man-on the totem pole by society, and in particular by school committees and administration. In addition, I would like high school math teachers to be mathematicians first, and education students second. I frankly am appalled at the student-teacher candidates I have been sent from a most prestigious university in Cambridge. These students in the mid-career program get an MAT in one year and often do not know their high school mathematics themselves. We have several times denied them placement in our school, but there is no question that they get that prestigious degrees in a year, and they do not know their math! Therefore I would want teachers all to have the equivalent of master's coursework in pure mathematics. What many education schools now offer teachers for their credits toward post graduate teaching are packaged quickie courses on discipline, or cooperative learning, often a series of tapes, but little pedagogical enrichment in pure mathematics which is necessary to pass on a love or the ability of doing good mathematics. Society should expect more of their teachers, and in return they should elevate them to positions of economic and professional power in their fields of expertise. It will only be then that mathematics in schools will again shine.

Question 5: I was a physics undergraduate at Radcliffe, who got a masters in pure math a Harvard and finished all the requirements toward a PhD except the research component. I then stopped working and raised my children and each time I contemplated a return to graduate study, I became discouraged by the non-support of family. At the same time I was also asked, "How did I know I'd ever be able to do mathematical research, that this was quite different from getting As in courses". Times were different, and I did not have the strength of will to continue without a lot of support and outside encouragement so I never returned to complete my graduate work. Instead, after ten or twelve years away from pure math I got my teacher certification and started high school teaching. The fact that I never completed my Ph.D. is one of the great disappointments in my life, and though I enjoy my teaching both in high school and in college (I teach and have taught adults at the Harvard Extension school and the Radcliffe Seminars) and enjoy working with students in their formative years, I miss the excitement and gratification of having worked in a research math/science field.

Question 6: a) My expectations of higher education for my students is that they receive the same fine teaching, guidance and support from faculty as I did in my undergraduate years at College. Despite much of the press that has written about sex discrimination in math, I personally never had experiences that were negative. Perhaps having some more women role models, who could have encouraged me to continue in my field while raising my family, would have helped, but honestly my professors were all very excellent and encouraging during my college and graduate school years. Perhaps the only adverse experience I had was that I was not hired as a TA as I was pregnant and as my husband was also a TA grad student too (nepotism at that lowly level!). T'was about forty years ago! I do want women and men to have equal opportunities to succeed, and I do hope in the zeal to protect women, we are careful to be fair to both sexes.

b) I have already discussed that prospective teachers must be better trained in their subject matter. Some are excellent mathematicians. Some should not be teachers without further training, and some should never be teachers.