Question 1: (a) They all must understand that every true statement in mathematics can be logically explained in terms of other true statements, and that nothing is ever true just because some higher authority decrees it to be so. This is the basic spirit of openness in any rational discourse, and if math education does not succeed in inculcating this spirit, then school education as a whole has failed miserably.

(b) Concomitant with (a), students must be able to distinguish clearly between what they know and what they don't know to be true. In mathematics and perhaps only in mathematics, this sharp distinction can be drawn and students should profit from it. They should not confuse a heuristic (but incomplete) argument with a conclusive proof. By the same token, once they have learned the proof of something in mathematics, they should be able to savor the satisfaction of unshakable conviction. The possible failure to achieve this goal in school math education is what worries me the most in the present math education reform.

(c) Students should appreciate that mathematics is a language of precision. As a language, the mastery of math must include fluency. They must therefore strive for this fluency. Thus certain basic techniques should always be available at their fingertips. In addition, the characteristic feature of precision of this language automatically excludes the kind of vagueness and ambiguity in everyday life. If students really understand this, then they would know that questions such as "What is best?" or "Is this fair?" (which are common in the mathematical problems of the current reform) have no place in any mathematical discussion until these loaded words have been precisely defined.

(d) The purpose of a math education lies not just in teaching students how to solve everyday problems (as some reformers would have us believe) but also in teaching them how to think precisely, logically and abstractly. The utilitarian aspect of the subject must also be tempered with an appreciation of its cultural aspects: its internal structure and its aesthetic appeal.

Question 2: The answer to 1 is a statement of the goals of a school math education. The school math curriculum should therefore focus on the attainment of these goals. One cannot hope to set forth in a few sentences how such a complicated task can be accomplished. The following are at best vague statements. An overall comment is that during the last two or three years of high school, the curriculum for students who expect to be quantitative majors in college should differ from that for students who don't in terms of the technical nature of the instruction. In general:

(a) Abstractions should be introduced early (in grade school), but in small doses. Students have to learn to handle abstract thinking, and the more they are shielded from abstractions in the early grades, the harder it would be for them to learn it later. For example, if fractions are taught properly (with the requisite amount of abstract reasoning), then 5th graders would already get to see the essence of mathematics at work.

(b) Informal reasoning for every mathematical statement should be given from the beginning. By the time of junior high, formal proofs should begin to make their appearance, if only in moderation. In the 11th and 12th grades, formal proofs should become routine (at least for students who will be quantitative majors in college).

(c) The applications in school mathematics should not overwhelm the curriculum, especially not those arising exclusively from everyday life. The power of mathematics is best demonstrated when it is seen to be essential for the formulation of far-reaching scientific principles rather than just the solution of picayune mundane problems.

(d) Mathematics is precise. For this reason, the possible ambiguity in the solution of real life problems must be clearly traced to the inherent ambiguity in the interpretation of the real life problems and not be blamed on mathematics itself. The sermon that "there is always more than one correct answer to a math problem", so often preached in the math education reform, should be laid to rest for good.

Question 3: (a) When students have achieved a basic understanding of (a)-(d) in 1. (b) When students have acquired the discipline of hard work through attempts at solving math problems.

Question 4: A school math teacher cannot be successful unless he or she knows mathematics and has mastered basic pedagogy. In principle, these two requirements are on equal footing. In practice, however, there are more people who know pedagogy than those who know mathematics. I have been in charge of the training of TAs in my department for over twenty years, and I have never ceased to observe repeated verifications of this fact. For this reason I would declare unequivocally that the most important attribute of a good math teacher is a good understanding of mathematics and love of the subject. Mathphobia in most people can be traced to a bad experience with a math teacher early in life, and the latter is often due to the feeling of insecurity on the part of the teacher resulting from a tenuous grasp of the subject matter.

In any math education reform, the first issue that has to be addressed is thus the qualification problem of the teaching staff. It would appear that in this country, the subject knowledge requirement of all teachers is less than adequate. All the more so in math. Until we can provide better teacher training for the prospective math teachers, massive teacher re-training programs for the working teachers, and great improvement in the working conditions of all teachers, the qualification problem will not go away and any discussion of math education reform would be futile.

Question 5: I had a terrible math education in grade school (in China) and consequently flunked every math course but one up to the 7th grade. Nothing was ever explained to me and everything was done by fiat. I felt I could never penetrate the secret code used in math. Then in the 7th grade, I had a great teacher. From the first day, he solved every problem in class by reasoning out loud. It then dawned on me that there was no secret code, just the kind ordinary reasoning that I could do myself. Soon after that, we started on proofs in Euclidean geometry, and this experience consolidated my feeling that math was a learnable subject. I have had little trouble after that.