Question 1: Recently Cal Moore, writing on educational reform in California, began as follows:

"while reforms are gradually taking hold, the majority of classrooms still rely on a traditional mathematics curriculum, that, as one cynical observer remarked, is largely composed of eight years of 15th century arithmetic, two years of 17th century algebra and one year of 3rd century B.C. geometry."

I would suggest that if students really have mastered these subjects they will have no difficulty with anything thrown at them in college.

Unfortunately, beguiled by ever fancier calculators and computers, teachers appear less and less able to produce students who are masters of these basic topics.

National leadership only muddies the waters. Listen to the NCTM (National Council of Teachers of Mathematics) on technology:

"...the ability of teachers to use the tools of technology to develop, enhance, and expand students' understanding of mathematics is crucial. These tools include computers, appropriate calculators (scientific, graphing, programmable, etc.), videodisks, CD-ROM, telecommunications networks by which to access and share real-time data, and other emerging educational technologies. Exploration of the perspectives these tools provide on a wide variety of topics is required by teachers."

All that's missing are strobe lights and a D.J. The simple, very old-fashioned lesson that "math class is hard," and requires hard work and lots of homework cannot help but get lost in the glitz of gimmicks.

Question 2: The answer here is tightly tied to the answer to question 1. Let me quote Cal Moore again: "while reforms are gradually taking hold, the majority of classrooms still rely on a traditional mathematics curriculum, that, as one cynical observer remarked, is largely composed of eight years of 15th century arithmetic, two years of 17th century algebra and one year of 3rd century B.C. geometry."

I firmly believe that this is the outline of a pretty good curriculum. If we consider a four year high school, I would suggest two years of algebra, one year of geometry, a semester of trigonometry, and one of analytic geometry.

The worst possible curriculum, as we have found out recently, will consist of "higher order thinking skills," "powerful mathematics," "earth algebra," and "mathematics for the 21st century." Each of these is a slogan without substance. The observation most easily forgotten is that you must crawl before you walk, and walk before you run.

Question 3: For anyone who teaches freshman calculus this is not a hard problem. When I sit with students in my office hours it becomes painfully obvious whether their high school education has worked well. If they can do neither algebra nor arithmetic, if they need their calculator to evaluate cos (0), then they have been miseducated. At the other end of thespectrum are the students who clearly have the basics under control and are able to learn the techniques and insights of calculus without being disabled by a poor mathematics background. Surprisingly, my recent classes have had large contingents from both of these groups, i.e. my As were the most numerous, followed by the Fs, the next most numerous grade.

Question 4: The most important attibutes of the best teachers are love of and facility with mathematics. These are what prospective teachers should gain from their college experience, where they should study mathematics, not education. They should then serve apprenticeships in schools where they would learn to teach under the supervision of master teachers.

There are real problems with the current preparation of teachers. Numerous fads and relativistic philosophy seem to be in the ascendancy in many colleges of education. Even if mathematics education departments in these colleges are comparatively free of these influences, nonetheless students will be afflicted with a variety of flawed courses and mixed signals. They should spend their college years learning mathematics.

Question 5: In high school, I was completely unaware of the possibility of being a mathematician. The only really exciting life I could imagine was suggested by the Sherlock Holmes novels. Here was a character, albeit fictional, whose entire career depended on adroit reasoning. Since such a career seemed to be pure fantasy, I decided to become a patent attorney because I was good at science and math, and found the law to be something involving some (I hoped) adroit reasoning.

In college at Oregon State, my life was transformed by Harry Goheen. He obviously loved mathematics with passionate enthusiasm, and he proselytized vigorously for math majors. He was able to rekindle in me the dream of a career built on the life of the mind. This began in his trigonometry course. By the end of his calculus course, I was a math major.