Question 1: Techniques; and I do not mean the use of computers as a substitute for techniques. I also care about the mastery of some of the important concepts. However, if the teachers are themselves not sufficiently familiar with these concepts, I would prefer that they make no effort toward teaching students their own version of what is and is not important. Teachers should have enough self confidence to say: "I don't know, I will try to find some expert sources."

The problem is that young students are very impressionable. Once they get a concept and value wrong, it is much more difficult to erase these and to replace them by the right ones.

Question 2: A variety of techniques in conjunction with their use in sciences such as physics, chemistry, biology. In terms of statistics and their use in the social sciences and humanities, I would concentrate on the mis-use of mathematics as described by short pamphlets such as: "How to lie with statistics", and instructive collections of problems such as: "50 challenging problems in probability", and "Chicken from Minsk".

Question 3: In my opinion: if the school consistently graduated students with high interests in the sciences (physics, chemistry, biology) and engineering (including computer applications), plus some in mathematics, then I would say that the math education at that school was successful.

Otherwise, this question is almost impossible to answer. There are too many plausible explanations. One needs to know the background of the students before they enter high school, and also the circumstances after the students have graduated. It is not so easy to conclude that in 3-4 years one has made up the deficiencies accumulated over the previous 15 or more years. In addition, one either has to project several decades down the road or has to wait several decades and draw conclusions without being able to factor in the variables in the decades after the students have left the school.

For example, in its entire 75 year history, a midwestern university high school graduated some 2,800 students. Prior to 1950, it did not have an extensive math program. At that point, the school recruited an outstanding high school math teacher who, with the help of a classical logician from the university math dept., started a version of the "New Math". It may have worked well, eventually, for that particular school. However, a large percentage of the students at that school were children of university faculty members; each teacher was assisted by one or two student teacher from the school of education at the university. At the same time, the various versions of the "new math" were notorious disasters in many other high schools that were not closely connected with a university.

Question 4: Most important: know a lot of basic techniques, content and the proper use of mathematics in the quantitative sciences and the improper use of mathematics in the more qualitative sciences.

Love of teaching and learning as a calling and then love of mathematics as a part of this.

Be aware of a variety of pedagogical approaches rather than being dogmatic about a particular approach. Keep in mind that teaching is not for the convenience of the teacher but for the future of the students. Know the background preparation of the students and their tentative goals beyond the immediate present.

Question 5: When I was an undergraduate lab assistant in a physics accelerator lab, I found that I could not understand the purpose of the nuclear physics experiment. I decided to read up on quantum mechanics in the library. To my frustration, I discovered that I could not understand the mathematics used in the texts. Not long after, I asked my best friend in college (a math major) what he was studying in mathematics. I was shocked to find that I could not read past the first few pages of his book with the esoteric title: Theory of Groups. Since my physics teachers had told me that I already overdosed on mathematics, I decided to ask my math prof about the propriety of beginning to study some pure math. (I had in mind the vague idea of spending most of my fourth college year to that end.) His answer was: "It is too late to begin studying pure math at the age of 19."

Two weeks later, I pigheadedly decided to graduate early and applied to the math department to try studying math full time. I did have a back-up. My chemistry teacher had mentioned during my freshmen year: "If you ever get tired of physics, come see us. Our door is always open."

Not long after, as warned by my science teachers, I was firmly seduced by "useless" mathematics. At the same time, I retained my interests in science and engineering and slowly tried to understand the more difficult and fascinating endeavors in the humanities. One does not have to be a genius to become a mathematician. Hard work and an open mind are, however, necessary. What I had found was that my early exposure to science and engineering enabled me to carry out useful collaborations and to discover interesting connections.