Question 1: I want a solid balanced program, based on problems, technique so that multistep problems can be done, and the structure and abstractness of mathematics so that an overall view can be achieved and students will know that the same mathematical ideas can be used in different areas and in different ways.

In the New Math period we tried to teach by concentrating on the structure of mathematics, feeling that computing and problem solving would be corollaries. For a few they were, but for most this was not the case. Then we got Back to the Basics, with a focus on computations and elementary techniques taught in isolation. This failed as well. Now we have the NCTM Standards [National Council of Teachers of Mathematics] which is trying to build a program on problems, with technical skills downplayed, proofs missing, and an overall picture of mathematics never displayed. This too will fail. Mathematics is like a stool with three legs, and all are needed for stability and usefulness.

I want students who can read and turn problems into mathematical equations or expressions. I want students to have grown up with a knowledge that mathematics can be understood, and while some things have to be memorized, there is a framework in which this can be done easily. For example, all students should understand geometric series, for they will buy things on time, a car or a house, and they should understand how interest payments are figured. However, they should not first see this in high school. In late elementary school, they should learn about the connection between fractions and decimals, and learn how to turn one into the other. For decimals this includes both approximations to the rational number, and repeating decimals. The same method which works to change a repeating decimal into a fraction should be seen a little later in summing the powers of 2 and 3, and then later the general geometric series, both finite and infinite. Unfortunately, the text books from 4th to 8th grades do this material very poorly if at all. Repeating decimals are often introduced via calculators at least two years before remainders are mentioned, and what is said about going from repeating decimals to rational numbers has to be read to be believed. To focus just on what high school graduates should understand about mathematics is too narrow, we need to be very concerned about how they get there as well.

I want students to be comfortable working with algebra, the geometry of triangles and circles, and all of those classical things which were mentioned in the letter signed by Polya and over 70 others which appeared in the American Mathematical Monthly and Mathematics Teacher in 1962, when they expressed reservations about the curriculum being developed then. I want students to be able to prove the Pythagorean theorem, for this is both surprising and useful. If students do not want to know why something which is both useful and surprising is true, then they are not getting an education.

For those who do not have ready access to the Monthly, vol 69 (1962) 189-192, there was a section labeled "Traditional" math. This section includes the following: "Elementary algebra, plane and solid geometry, trigonometry, analytic geometry and the calculus are still fundamental, as they were 50 or 100 years ago: future users of mathematics must learn all these subjects whether they are preparing to become mathematicians, physical scientists, social scientists or engineers, and all these subjects can offer cultural values to the general students." They go on to say that what has been wrong is not the subjects but that so much of it was treated in isolation from other domains of knowledge and inquiry, especially from the physical sciences and even other parts of mathematics.

If we focus on just one thing which we say we want most, we will fail, for mathematics education is too complicated to be put into a one line format.

Question 2: I do not feel that all students should take the same curriculum in high school. No other industrial country has the same program for all students in high school. Students should have a choice of which type of mathematics course to take after, say, 10th grade, and it should be their choice rather than something forced on them by someone else. However, all of the programs should cover significant mathematics well enough so students can use it to do problems which are not exactly like ones they have seen worked before. There should be enough about the nature of mathematics so that they can understand what was written in newspapers about the solution of Fermat's conjecture, and know that this is important even if it does not have immediate use in their lives. This is asking a lot, for even a very smart person like Marilyn vos Savant did not understand what was happening with respect to Fermat's conjecture. This illustrates why schooling is necessary. She felt she knew enough to be able to comment on this, and claim that mathematicians did not know what they were doing. Students should learn enough about the history of mathematics and its relationship with science to see what hard work was involved in developing this, and realize that the same type of hard work is needed now if more developments are to happen.

Question 3: We have many problems in our schools, and mathematics education is only one. However, many of these problems are related, and I do not think we can solve one in isolation. Thus rather than try to say when I think the whole problem of school mathematics has been fixed and is working well, I think it is sufficient to look at one problem there and see when it has been solved.

The one which is easiest to look at is the problem students taking calculus have with algebra. For far too many, their lack of skills in algebra makes it very hard--and, for many, impossible--to learn calculus. Some have tried to attack this problem by redefining calculus, just as they have tried to attack the problem of poor arithmetical skills causing problems in algebra by changing what algebra is. Neither of these will work, and we need to go back to fundamentals and solve these problems directly. Elementary school is the most important.

In the book The Shopping Mall High School (Houghton Mifflin, 1985) the author of Chapter 2, Arthur Powell, describes a treaty between many teachers and many of their students. In short, it is: I will not force you to learn anything in this class if you do not cause any trouble. While this is not true universally, I am afraid it is an accurate description of far too many classes. Many calculus students have told me that they wish they had studied in high school, but no one else did, so they did not either. Some of this talk is for show, but far too much of it is accurate. When students no longer say this, then significant progress will have been made.

Question 4: A solid knowledge of mathematics is needed. This must not only include the mathematics being taught, but mathematics which is taught for the next few years, for how something is taught should partly be determined by the desire to point students in the right direction to learn later material. For elementary school teachers this means that they not only need to know elementary school mathematics well, they need to understand and be able to use algebra and know geometry well. They also need to know how mathematics is used in other fields and in the day to day life of people, both in everyday life and something about how it is used in some professions. For example, the Pythagorean theorem and its converse need to be known, a proof should be understood well enough so that a question from a curious child about why it is true can be answered, and its use in the building trade to square corners should be known. I mention this for elementary school teachers, for that is the critical period where mathematics must be taught well, and far too frequently it is not.

The solid knowledge of mathematics through calculus, linear algebra, some number theory and combinatorics, and some probability theory and statistics is all needed by high school teachers, as is a good knowledge of geometry. It is also very useful to know a fair amount about the history of mathematics. A knowledge of physics is also very important, for much of mathematics came from a desire to understand nature, and physics is the place where most of this happened. High school mathematics teachers should like mathematics, want to discuss it with others, and continue to learn mathematics. Reading a magazine like Quantum is an excellent way to do some of this, but book reading is also important. About teaching, teachers should care about their students, listen to them and be able to respond to their spoken and unspoken questions, and help students want to learn mathematics and to learn it. I have lived long enough to have seen a number of different teaching styles pushed as the only way to teach well, so do not feel that there is one best way to teach which can be taught to all and for all.

Question 5: The story my mother tells is of me in a high chair doing a follow-the-dot puzzle and asking for a calendar since I had finished with the clock. I always liked mathematics, and in high school met a second cousin who had MA degrees in mathematics and physics who told me that it was possible to do mathematics. Before then I had wanted to be a physicist.