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I'm teaching mathematics at a high school and I am offered the opportunity to get extra lessons, namely: I can provide an optional course about some topic of mathematics. The optional course will take two lessons (each 45 minutes) a week and last a whole semester.

The mathematical skills of the pupils will vary a lot: Usually, they will have covered one or two the three topics "vector geometry", "differentiation" or "integration". I myself am mostly interested in linear algebra and theoretical physics (quantum physics).

I'm having a hard time to come up with something interesting I know a lot about to teach them. Obviously, I can also read a bit about something not in my area, because I doubt I can show them any proper theoretical physics because a) it won't count as a maths course and b) not all of them know integration and none know complex numbers yet.

Does anyone have a suggestion what I could do? A friend suggested to do number theory but I'm not sure if that is appealing to high school pupils (I never cared for it) plus I still don't know anything about it.

//edit: It should also not be something that is taught in the standard curriculum, e.g. not complex numbers.

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We had something like that when I was in high school. It was 40 minutes a week and we studied Axiomatic Geometry. I benefited a lot from this course! Now, looking back, I think I understand why.

First, we were already familiar with the basic theorems of geometry so it wasn't intimidating.

Second, it was a great opportunity to learn some history of mathematics, to see where and how it began.

Third, it really prepared (some of) us for the university courses we later took. So when I got to the university, I was already familiar with axiomatic approaches, definitions, lemmas, theorems, the concept of formal proof (from the axioms) etc.

Overall, it was fun!

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  • $\begingroup$ I loved axiomatic geometry in university, however there is another course at the high school about it already. $\endgroup$ – Huy Sep 20 '14 at 19:12
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I highly recommend introducing the students to a survey of discrete mathematics: sets, logic, counting techniques, probability, graphs, all that good stuff.

Most likely the students will not have encountered such a topic before, and will be pleased with the break from the monotonous calculations of algebra and calculus. Even the students who are not so hot about mathematics will benefit, since the aforementioned topics are usually covered in a course in university along the lines of Mathematics for the Liberal Arts.

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  • $\begingroup$ I've only come across discrete maths in a course about algorithms and complexity at university, covering the very basics of graph theory. What applications for a high school student would you see in this? Any mind-blowing theorems for them? $\endgroup$ – Huy Sep 20 '14 at 19:14
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  • Treat it like a math circle, where you propose an 'accessible mystery.'
  • You don't need to care about how much content you 'cover', so you can focus on the students' mathematical thinking, and help them to uncover mathematical concepts.
  • Pick a topic you like, so you'll have passion for it.
  • Be aware that your students will struggle with actually thinking mathematically, instead of the usual method of following procedures they've been shown. (Sometimes 'good students' struggle more, because they're used to it coming easily.)

Possible topics:

  • Non-euclidean geometry might be cool. I don't think you'd need much research to be able to mentor the group well in this.
  • If you do want to address complex numbers (unclear to me from your post), a good starting question might be "What is i to the i?"
  • Fold a long strip in half repeatedly, creating a dragon curve. Describe it mathematically. Go on from there...

Check out this site for more ideas: http://www.mathcircles.org/content/math-circle-problem-collection

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  • $\begingroup$ I've done some non-euclidean geometry in university but I don't know how I would approach it. We came across it with an axiomatic approach and there is already a course about axiomatic geometry, so I'm scared they would intersect. I'll check out your site though, thanks! $\endgroup$ – Huy Sep 20 '14 at 19:16
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A fun topic that could be covered at the high school and would introduce students to more advanced ways of thinking is Combinatorial Game Theory. You could use Volume 1 of Berlekamp, Conway, and Guy's Winning Ways for your Mathematical Plays as the source for your material. You can let the students discover strategies for various games, and depending on your judgment, you can also consider discussing how one proves that the optimal strategies are just that.

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As a topologist myself, I am somewhat biased, but given such an opportunity I would teach a course on low-dimensional topology, using a book such as Topology Now! or Introduction to Topology: Pure and Applied. If you are willing to do something a bit more focussed, one could use something like The Knot Book.

Topology Now! in particular does not require a whole lot of background, and whatever background is needed could be covered as an aside during one of the weeks (since it's an optional course it presumably doesn't matter too much how far you get into the chosen book.). The reason I recommend this, apart from the aforementioned potential bias towards my own field, is that topology/knot theory are really quite different from the types of math HS students usually see; as a result, such a course would reveal to them some unexpected places where math plays a role, and also possibly level the playing field a bit for your students who, as you said, might have different mathematical backgrounds.

Here is a short review of two of the books I mentioned: http://www.jstor.org/stable/40391111

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An elementary introduction to Graph Theory might not be a bad idea. It has at least the following pros:

  • unlike the other math classes
  • will likely never see this in undergraduate coursework unless if they are going into Computer Science or Mathematics (maybe Electrical Engineering)
  • ample topics and applications associated with graphs
    • the internet
    • social networks
    • map software that gives "optimal" routes
    • map coloring
    • knights tour
    • traveling salesman
  • nice puzzles
    • Konigsberg bridges
    • you can also make up your puzzles!
  • word games like "get from paper to sketch in as few steps as possible by changing one letter at a time to construct a new valid word" (can this be done?)
  • exploration of board games
    • Ticket To Ride
  • is accessible to almost all math skill levels (I've taught Graph Theory units at the college level and my experience has been that no matter how "weak" the student may be with the traditional math curriculum, a basic introduction to Graph Theory is still within their reach)
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