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:
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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.
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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.
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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?)
Best Answer
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.