# Topic for a high school-level math elective?

I'm looking for ideas for a 15-hour mathematical enrichment course in a Chinese high school. What (fairly) elementary subject would you suggest as a topic for such a course?

Background/considerations:

• My students are generally quite good at math, but many of them have not been exposed to rigorous or abstract mathematical reasoning. A good topic would be one that would not be impossibly difficult for students who've never written or read proofs in English.

• I've taught this class three times before. (Part of the reason that I'm posting this is that I've used up all my ideas!) The first semester I taught an introductory number theory class (which meandered its way towards a proof of quadratic reciprocity, though I think this was ultimately too advanced/abstract for some of the students). The second semester I taught basic graph theory and applications (with a focus on planarity and coloring). The third semester I taught a class on the Rubik's Cube.

• The students' math backgrounds are quite varied: some of them participate in contest math competitions, and so are familiar with IMO-style techniques, but many are not. Some of them may know some calculus, but I can't count on it. All of them are very good at what in America is sometimes termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They know what a binomial coefficient is.

So, any ideas? Ideally, I'd like to find something a little "sexy" (like the Rubik's Cube) -- attempts to motivate number theory via cryptography seemed to fall on deaf ears, but being able to "see" group theory on the cube was quite popular.

(Responses especially welcome from people who grew up in the PRC -- any mathematical topics you wish had been covered in the high school curriculum?)

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I believe this should be set Community Wiki by some moderator. –  Asaf Karagila Jan 8 '11 at 13:24
My suggestion: (elementary) geometry of complex numbers, the complex exponential function, and the fundamental theorem of algebra. –  Matt Calhoun Jan 8 '11 at 16:07

I grew up in PR.China, and was quite disappointing with the pre-university education in mathematics. I am very happy to see one educator like you posting such a question here.

Combinatorics, graph theory and number theory, in my opinion, are proper fields you can choose materials from. By choosing some topics relating to "big theorems" such as Fermat's last theorem (of course in relatively naive ways) can surly attract young students.

I think this could be done topic by topic, instead of stucking in only one small field. I believe one major problem in mathematical education in China is that there are too many restrictions on different branches. There are too many questions such as "what field does this problem belongs to?"

A book to recommend is "proofs from the book" written by Martin Aigner and Günter M. Ziegler (with illustrations by Karl H. Hofmann). Although this is wriiten as a graduate level book. One can find materials suiatble for high school students. More importanly, it can greatly enhance the students' taste in modern mathematics.

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Written by Martin Aigner and Günter Ziegler, to be precise (with illustrations by Hofmann). –  Hans Lundmark Jan 8 '11 at 17:06
corrected, thanks. –  Xiaochuan Jan 9 '11 at 11:24
Great info. Thanks! –  Jon Jan 10 '11 at 4:30

Except for the topics that were already mentioned (mostly those that come from combinatorics, which can be explained relatively easily to students without advanced mathematical background, and also contain many interesting puzzles that can be given to them for solving), I would suggest some basic probability.

You could explain different everyday counter-intuitive problems such as the Monty Hall Problem or the Boy born on a Tuesday problem, and this topic doesn't require much advanced background.

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Perhaps some non-Euclidean Geometry? What's more fun than pulling the rug out from under everything the students thought they knew about geometry? :)

Marta Sved's Journey into Geometries provides a readable (if pricey) take on the subject, in the style (and with the characters) of Lewis Carroll's "Alice in Wonderland". When I first read through the book many years ago, I wanted to share it with students, though I didn't have the opportunity. The entire book may be too much for kids alone, but filtered through a teacher, some notions (such as the "Power of a Point" relative to a circle) can be pretty accessible, as I recall ... but my memory of the details is quite sketchy. (I seem to have lost my copy of the book.)

I believe there are hyperbolic add-ons to software like Geometer's Sketchpad that facilitate playing with circle inversions and such, if that's an option. (No doubt there are many applets on-line as well.)

If the students are in fact very good at (Euclidean) trigonometry, they may find non-Euclidean trig interesting. If nothing else, students may gain an appreciation for the convenience of our (locally) Euclidean universe and its straightforward way of computing the length of the hypotenuse. (On the other hand, it's hard to beat the formula for the area of a hyperbolic triangle.)

And many Escher woodcuts are intriguing visualizations of (approximations of) tesselations of the hyperbolic plane.

In general, I'm in favor of any enrichment exercise that keeps geometry from being "that weird class between Algebra I and Algebra II" that students tend to forget all about.

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I think that in the long term the most important thing you have to do is to stimulate their mathematical muscles.

If you want to show them mathematical rigor (which is a good idea) I suggest some propositional calculus, or predicate calculus. If you want you could solve some riddles using truth tables or by inference rules (an example in my answer here: Implication of three statements)

You can show the importance of rigor by giving examples to how "All horses are black" or "For every natural number $n$ it holds $n=n+1$", or find some other example which is simple for them.

Perhaps you can show them some infinite structures and how infinity behaves differently, start with Hilbert's hotel, and then show them how you can "compress" all the integers into natural numbers, or assign a unique natural number to every rational number - with room to spare.

Whatever you choose doing, I think that the most important thing is not to get too deep into technical details and proofs, and keep a bit of the mystery to get them search for it themselves.

I just remembered something you might find helpful, http://mathoverflow.net/questions/47214/how-to-present-mathematics-to-non-mathematicians this MO thread which was opened by yours truly, it deals with a different situation altogether (I was giving a 10 minutes presentation to non-mathematicians) but it has a nice list of topics that you can take and further analyze and deconstruct in your time.

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I'm not sure how well this would work in practice, but I've always been interested in introducing mathematical rigor through an abstract algebra course, rather than through analysis. As a motivating question, you could have the problem of squaring the circle (for example), though this would involve going into a lot of theory pretty quickly. (It might also be more appropriate in the west, where squaring the circle and trisecting the angle have been problems of interest since antiquity.)

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Yes, I think that some sort of an "Intro to Groups" course would be a good change of pace for mathematically mature high school students. Plus it would show them that math isn't just about numbers and computation. –  jericson Jan 8 '11 at 22:39

As a mathematically-inclined student currently in high school (although not in China, so perhaps my opinion is less relevant than it would appear to be), two subjects I would suggest are basic logic and reasoning (usually introduced in the US as part of geometry, which makes it seem boring and less relevant to "real life" reasoning), and basic probability.

As to how to make these "sexy", I honestly don't know for the first one. Perhaps you could combine it with some topics from computer science or, more concretely, programming computers.

Probability is relatively easy to motivate, since it has applications to a very large number of areas outside of mathematics.

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It might be helpful for you to get ahold of some of what I've heard called "liberal arts math" textbooks—more or less, discrete math books aimed at college students not intending to pursue much further study in math. The two that I'd suggest offhand are COMAP's For All Practical Purposes (published by W. H. Freeman) and The Heart of Mathematics by Ed Burger (published by Wiley). These and similar books have a number of topics that are presented in ways designed to draw in students of varying backgrounds. Many of these topics are not typically in U.S. secondary school curricula and can readily be extended from a "liberal arts math" level to more rigorous math-major style mathematics (e.g. voting theory, fair division).

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