A Better Mathematics Curriculum Essay

3641 words - 15 pages

1 My MotivationAll these ideas have been churning around in my head for years. I've finally written down this draft containing some of them. Although it is very rough, and I'm sure I've left out many obvious things, I would appreciate comments.The curriculum I'm proposing is aimed at high school and below, although obviously some of the ideas make sense in a university setting.Just to show that I'm not completely ignorant of the situation, here is my background: I have a B.S. in mathematics from Caltech, and a Ph.D. in mathematics from Stanford. I have taught courses in mathematics and computer science at various universities ranging from Stanford to junior colleges. I have done a great deal of volunteer tutoring of mathematics, for kids and adults who have a great deal of difficulty, and for kids who are far smarter than I am, and have competed on various United States Olympiad math teams. I also did a post-doc at Stanford in electrical engineering and have worked in industry as a software engineer for 20 years.2 The Problem TodayI think the problem with mathematics education at the university level in this country is that it's generally taught by and aimed at mathematicians. This trickles down to the primary and secondary schools, since the committees that determine the curricula are usually packed with university-level mathematicians.I was trained as a professional mathematician, and in my non-mathematician, non-engineering life, I have never really needed to solve a quadratic equation either.You would think that in careers that use mathematics heavily - for chemists, physicists, engineers, and computer scientists - at least they are learning the right stuff, but I don't think that's the case. When I took third-year physics as a junior I had to work with Fourier series using actual sines and cosines. I had, of course, learned a great deal about Fourier series in my math courses, but all at a highly theoretical level. We didn't use sines and cosines; we used "complete sets of orthonormal functions on a measurable space" or something. We learned about the weird convergence properties of the functions that were on the edge of not having a Fourier expansion. The bottom line was that I had to teach myself to do it in the "usual case" with sines and cosines applied to reasonably well-behaved functions.]Later in life, I've run into dozens of similar cases, where I had learned the abstract, theoretical theorems, but had never tried to apply them to real problems. If you read Concrete Mathematics by Knuth, Graham, and Patashnik, Knuth states in the preface that the reason the book and the course based on it were written and taught is that there were large areas of mathematics he had never seen taught and that he wished he had known in order to do his work in computer science.In many universities, in fact, engineering departments offer their own math courses since their students are unable to solve engineering problems with the tools they learn from the...

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