Question 1: High school graduates should understand that the aim and function of mathematics is much more than obtaining numerical answers to questions by memorized rules of computation and use of computers or calculators. The fundamental processes of mathematics include (i) abstraction and representation (by symbols, formulas, diagrams, figures), (ii) symbolic transformations (which include computation and deduction), (iii) application and comparison (e.g., of model and "real world"). These processes need not have been discussed and identified when they were students, but they should all have been experienced by the students, and used in problem-solving. This experience should inform the students' ideas about what mathematics is and "how it works".

Question 2: (a) In mathematics we have special ways of using language which enable us to formulate and communicate our ideas precisely. Students should learn to use this language comfortably to express their own ideas, as well as to penetrate the ideas of others. One feature of the language is the free and frequent use of letters as temporary, short names for things about which one seeks, and then communicates, information; these include sets of things in which one is interested. Students should appreciate that formulas and equations are not simply mathematical artifacts to be manipulated, but are linguistic elements used to express ideas about the objects represented by their letters.

(b) Students should realize that the search for a solution to a problem need not follow any rules or restrictions, but that the way we present the solution after it is found often proceeds along a very different path that is governed by rules. They should be practiced in making guesses in the search for a solution, and trained to check each guess with the expectation that most of them will be wrong. They should understand that when wrong guesses are examined carefully, they often lead to the formulation of better guesses.

(c) Students should be encouraged to pose questions of their own, especially to extend a result they have obtained by solving a question that has been put to them. They should experience the satisfaction of solving a problem by devising a method that has not been laid out in advance. They should learn to cooperate in group efforts to solve a problem. Just as in sports, cooperation and competition can be fitted together in problem solving, adding zest to the mathematical experience.

Question 3: In mathematics education as in mathematics practice, we feel that things are going well when we attain an objective by overcoming substantial difficulties. In both activities, success leads us to set new and more difficult objectives, immersing us again in feeling that things are not working so well. An individual teacher, a school, a community, a region, a country..., each of these should set goals that are difficult but attainable, and we should realize that dreams that now seem unattainable may yet be realizable step by step. Mathematicians and teachers must join with others in the community to help all students achieve their potential. When things are not working well, keep trying. When they are working well, set higher goals.

Question 4: By way of mathematical background, it is most important that high school teachers have a good understanding of the parts of algebra, geometry, and analysis into which high school math will flow when students continue to study math in college. They must also be familiar with a wide spectrum of applications of math that can be made in high school science courses, and in non-academic fields where many high school students may seek jobs after graduation, including the use of computers.

By way of attitude, teachers should enjoy mathematics, let their students see this, and help them to achieve it for themselves. This is much more important than ensuring that the students learn some body of facts in class, for if they truly enjoy working with mathematics they will go on learning long after leaving the class. Teachers must be sensitive to the fact that students are only learning to use language in the same way as the teachers themselves. Hence, a student may be trying to express a correct idea in words that sounds incorrect to the teacher; and if there is a wrong idea in the student's mind, it may have arisen from a natural misinterpretation of the teacher's words. By way of pedagogical methods, teachers should elevate to first place the use of praise--not only for correct answers, but also for brave guesses even when they turn out wrong.

Question 5: As a schoolboy, I saw that I could be certain, entirely on my own, that I had arrived at a problem solution that was perfectly correct--in mathematics. This set math apart from all my other studies. When teachers and family gave praise for such successes, I was encouraged to go further. But it was only the marvelous depth of mathematics that suddenly opened before me when the deductive method appeared as the core of upper-division college courses, that was the decisive component in getting me to consider seriously a career built around mathematics.