There are several problems with teaching Algebra II today. The first is the complete lack of motivation of virtually everything that is taught. The second is the lack of an easy-to-use algebraic manipulation tool -- e.g., Maxima/Maple/Mathematica -- which can automate tedious algebraic manipulations and graphing. Solving the motivation problem is going to be extremely difficult, because very few people have ever had to work out what the proper motivation for each element should be. The usual motivation -- following the historical order of development -- can work, but these tend to get side-tracked with stories about Cardano instead of stories about what motivated Cardano to take the steps he took. What is missing is someone of the caliber of Richard Feynman motivating the development of each step. One powerful modern motivation for analytic geometry is computer games & computer graphics. One possible motivation for Algebra II in high school might be the development of the computer graphics for a computer game. For example, so-called "homogeneous coordinates" never made any sense to me until I saw how elegantly they handled many problems in computer graphics. Historically, the "core" of algebra goes from solving linear equations to solving quadratic equations, to solving cubic & quartic equations & non-solving quintics. The nice thing about this sequence is that we first build N, then Z, then Q, then quadratic extensions of Q (including the complex numbers), then cubic extensions of Q (including cubic roots of unity), and trigonometric solutions. If we have enough time, we can do quartic equations and then get into the unsolvability of the quintic -- e.g., Klein's icosahedron. In one "smooth" progression, we can quickly bring in an incredible amount of mathematics, including rational functions, complex numbers, circular & hyperbolic trigonometry, group theory. So long as we can automate much of the drudgery with computer tools, and use the computer to do pretty animated pictures & diagrams, I think it should be possible to teach many people who hate math all of this knowledge. It isn't necessary that most of them be able to reproduce it, but if they can follow the pretty pictures, they can see that there is beauty there, and can come back later to do it more rigorously. Probably the worst thing that ever happened to high school math in the 20th century was the "rigor mortis" that was introduced in the middle of the century. Rigor is fine for professional mathematicians, but most mathematicians I know don't start with a rigorous argument. They try to "see" a path to a solution, and then fix up the loose ends later. This is how math should also be taught: basic insight, followed by details & fixups. --- If you haven't tried the Khan Academy math courses, you really should. The Khan Academy algebra & geometry courses are really quite good, and one heck of a lot better than 99% of the textbooks & coursework available to high school students today. https://www.khanacademy.org/ --- I've been trying to play a very small part in this process with some web pages. Here is one: Solution of Cubic Equation with Three Real Roots http://home.pipeline.com/~hbaker1/cubic3realroots.htm I'm also working on another one for solving the general cubic with complex coefficients: What Affine Day to Solve a Cubic (I posted the main portion of this page to math-fun a number of months ago.) I hope to have this page up in a month or so. At 07:41 PM 11/14/2013, Dan Asimov wrote:
Did anyone happen to read the article "Wrong Answer: The Case Against Algebra II" by Nicholson Baker, in the Sept. '13 Harper's magazine?
I'm curious what other math-fungi think of it. If you'd like a copy just let me know and I'll send you one.
--Dan