# Short Course Suggestions For High School Students

I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are familiar with single variable calculus and very basic linear algebra over the reals. The teaching format will be two hours per day, six days per week and two weeks in total. Suggestions for one week courses are also welcome.

There are two things I want about this course. First, it should have a direction and a final goal. So it shouldn't be based on isolated Olympiad-type problems. Second, it should introduce at least one new concept or object which is not a part of the high school curriculum.

For instance, classification of frieze patterns is a good topic. There is a clear goal and one needs to introduce the concept of a group which is new for high school students. Any other suggestions?

• In case you decide to go with something that involves ruler and compass constructability and/or the theory of equations (by which I mean what was a standard undergraduate course from the late 1800s until its disappearance in the 1950s), you may find the following manuscript of use: pballew.net/Constructable_17gon.pdf I wrote this for situations such as you find yourself in, more as a secondary reference than as a primary reference. – Dave L Renfro Dec 21 '11 at 20:30
• Have you seen Etingof's notes on group theory? – B. Bischof Dec 22 '11 at 4:58
• I have a series of lectures given to supplement a low-level freshman university course that cover RSA, P/NP, Turing machines, diagonalization (a la Cantor, Turing, and Godel), random walks, and the central limit theorem. If you want copies, email me (my contact info is on my MO profile). – Steve Huntsman Dec 22 '11 at 16:48
• Include some homework exercises with your lectures! – KConrad Dec 23 '11 at 8:55

If I may forgiven for self-promotion, you might examine How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra (Cambridge University Press, 2011). All of its topics are accessible to high-school students, but all fall outside the high-school curriculum. See also howtofoldit.org for some (not yet well-organized) supplementary material.

You have not told us where the students are coming from but it is pretty safe to say that these days no high school students (with exception of few countries on the world) are exposed to any meaningful Geometry course.

How about teaching them some real old fashion synthetic (Euclidean/Lobachevsky) geometry course based let say on Kiselev's classic

http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202

with possible excursion into Projective geometry.

There is no more natural place to introduce the concept of groups (actions on the sets) than in Geometry (composition of isometric transformations). There is no more natural place to introduce them to the concept of measure. It is very easy to involve hard combinatorial problems and many other things. Finally, set theory is all over the geometry and axiomatic method rules.

Many of the most challenging "Olympic problems" are geometric in its nature.

Best,

Predrag

If you can teach game theory, that could be good. It's bread and butter for mathematical economics and political science (even ecologists learn it now) -- I think the subject illustrates the point that math is not limited in application to situations which involve numbers. In addition to being useful, it's very elementary to solve games (although the fundamental fact that mixed strategy Nash equilibria exist requires topology to prove, it doesn't provide an algorithm for finding them -- actually solving games is more combinatorial). Proving that sets of strategies are/are not Nash equilibria can introduce students to the concept of a formal mathematical proof in a setting which I think is straightforward.

Unfortunately, I can't think of a textbook that would be good, but maybe someone else knows one.

• Maybe Game Theory and Strategy by Philip Straffin, Jr.? It is an MAA Textbook. A reviewer said, "The only mathematical background necessary is that found in the college-track high-school curriculum." – Joseph O'Rourke Dec 22 '11 at 2:51
• Winning Ways is fantastic for combinatorial number theory, because everything is built up through games which anyone can play, and numbers only come in as a notation to keep track of who is ahead in the game. Plus the "numbers" include surreal numbers like stars and arrows. I remember seeing this during a math summer camp when I was 15 and being shocked that something that wasn't a number (like "double up star") could be used to count things, and that those numbers could be combined (added) in a meaningful way. – Zack Wolske Dec 22 '11 at 6:07

My first suggestion would be a course on set theory. Starting with naive set theory, you examine the diagonal argument, paradoxes, and early developments. Then you use that to motivate axiomatic set theory (perhaps ZF), derive Peano Postulates, prove Cantor-Schroeder-Bernstein, survey cardinal arithmetic.

If they've seen computational side of calculus, another idea could be to do introductory analysis course. Assuming little but rational numbers, you could construct real numbers, show their uncountability; then do the calculus they were taught, proving everything on your way.

Other suggestion with which, I feel, one cannot go wrong, is elementary number theory. I would stress the prime number theory, proving Bertrand's Postulate and stating prime number theorem.

Full disclosure: I'm currently a high school student.

• I can't help stating the fact that neither of the three suggested courses would improve my interest in mathematics in the long run. Set theory fascinates people when they first hear of it (I too had such a phase during my school time), but at some point tends to lose them and leave nothing behind except for an impression that mathematics is about playing around with virtual infinities beyound any hope of real understanding or intuition. What, for instance, does Cantor-Schröder-Bernstein mean? I also don't consider it a good idea to give a more technical and axiomatic take on a subject ... – darij grinberg Dec 22 '11 at 1:39
• ... already overstressed ad nauseam in school, such an analysis. In my circles (German IMO training 2004-2006) the predominant feeling towards (single-variable) calculus was annoyed contempt; differentiating and integrating was something to be left to lesser beings (applied mathematicians, schoolteachers), whereas topological stuff like continuity and the construction of the reals was considered reasonable but boring and technical. Only complex analysis evoked better feelings. This probably has to do with the fact that in school, the only analysis taught is trivial and boring analysis. ... – darij grinberg Dec 22 '11 at 1:42
• ... If you want to change something about this, I think the better idea would be to introduce some of the more surprising and fresh material (complex analysis, $p$-adic analysis, nonstandard analysis - handle with care -, even some of the nicer numerical analysis), rather than just to do the standard 1-dimensional real calculus with more attention to proofs and details. – darij grinberg Dec 22 '11 at 1:44
• As for elementary number theory, it is a good proposal, but the Prime Number Theorem is a problematic matter: just stating it doesn't help a lot; proving it might be too much for the course. I'd better go with quadratic residues and some of the more interesting modular arithmetic. – darij grinberg Dec 22 '11 at 1:48
• My feeling about set theorists is that many of them have been brought to cardinal arithmetic etc. through its applications (forcing, decidability and computability questions etc.), rather than doing it for its own sake. A course taking these real applications (instead of the faux naturality of studying infinite sets) could be very interesting. – darij grinberg Dec 22 '11 at 16:17

Serge Lang's book "Math Talks for Undergraduates" (Springer, 1999) has quite a few topics which will work for anyone with some calculus. Topics include symmetric polynomials, approximation theorems in analysis, prime numbers, and the abc conjecture.

I believe (and I may have some of the details wrong) Mark Sapir had at one time a curriculum for fourth graders that involved combinatorics on infinite words. I do not know if he still has it, or how adaptable it is, but it introduced the Thue-Morse sequence and (I believe) had some applications, such as (non-FIDE) unending chess, analysis of certain dynamical systems, and so on. Since I may be mistaken as to the details and availability, I suggest asking Mark Sapir or rolling your own.

I realized recently that you can do something really cool with good students after they learn the standard forms for conic sections: you can compute the compactifications of their moduli spaces. I gave an undergraduate talk based on this, and I think it went really well. You have to wave your hands a bit and you might not want to use the word compactification. It is pretty obvious how to draw the uncompactified spaces of conic sections centered at the origins, but something really cool happens when you approach the boundary. I think this could be stretched out a bit longer than an hour and you could probably do several nice lectures on it, one for each different moduli space.

Let me know if you come up with any new low level examples for the moduli spaces. Mine were: -Triangles in the plane, -circles centered at the origin, -circles in the plane, -ellipses centered at the origin, -hyperbolas centered at the origin.

You could also look at how the discriminant is a function on the moduli space.

I think a course about homogeneous linear recurrence relations with constant coefficients should be manageable. The simplest nontrivial example is probably the Fibonacci recurrence $$F_{n+2} = F_{n+1} + F_n.$$

A large supply of nontrivial accessible examples is given by counting walks on graphs or, roughly equivalently, words in regular languages (e.g. the language of all words not containing a particular word $w$). When the characteristic polynomial has distinct roots, the solutions are given by powers of the roots, and this is a very nice example of how using a non-obvious basis for a vector space (the vector space of all solutions) can clarify a situation, and also a fairly concrete example of how complex numbers can naturally occur in answers to real questions (if the characteristic polynomial has complex roots).

The general case is somewhat difficult to explain directly, but can be described using any of the following approaches, roughly in increasing order of abstraction:

• partial fraction decomposition of a generating function,
• factorization of a polynomial in the shift operator $S(f_n) = f_{n+1}$,
• Jordan normal form of a companion matrix.

The second approach allows the clearest analogy to the case of homogeneous linear ODEs with constant coefficients if the students are familiar with those.

Of course to cut down on the abstraction it's probably best to focus on examples, and I think the students will be pleasantly surprised at how many difficult-looking combinatorial questions reduce to the counting of words in regular languages, which turns out to be relatively easy.

There is also a cute connection to Pisot numbers, e.g. it is not obvious why the powers of $2 + \sqrt{3}$ should rapidly approach integers until you realize that $$(2 + \sqrt{3})^n + (2 - \sqrt{3})^n$$

is a sequence of integers satisfying a linear recurrence with integer coefficients and that $|2 - \sqrt{3}| < 1$; moreover, this sequence counts the number of closed walks of length $n$ on the multigraph with adjacency matrix $\left[ \begin{array}{cc} 2 & 1 \\\ 3 & 2 \end{array} \right]$ so has a direct combinatorial interpretation as well.

The closest thing I know to a complete reference for this material is Chapter 4 of Stanley's Enumerative Combinatorics Vol. I; section A.I.4 of Flajolet and Sedgewick's Analytic Combinatorics may also be useful.

• see also "Recursion sequences" by Markushevich; this book is written for kids – Anton Petrunin Jan 1 '12 at 1:36

Chip-firing, rotor-routing and cycle-popping can be understood by anyone with or without combinatorial background, and provide new insights on lots of old combinatorial problems (counting Eulerian cycles and spanning trees, for instance). Here are some more basic facts, and here are some newer results. While the papers linked are probably too concise and too scholarly to be understood by students directly, it shouldn't be that difficult to make the results accessible for school students by writing them down in a more expository manner. (Needless to say, this would actually add a lot of value.) There are some notions from algebra used (group, group action, monoid, determinant), but (except for some linear algebra, which also can be avoided if so desired) mostly just the language is being used, not any nontrivial theorems.

Gröbner bases and elimination theory are another good field, but I don't have a good elementary reference for this. The question how to solve a system of polynomial equations in general is a natural one and a good student should have asked himself this question at least once. Unfortunately the answer is never given even in university lectures. Algebraic geometry is not an answer.

Now that we are talking about solving equations, I remember Vladimir Arnold having written a school-level (well, something he considered school level, referring to Russian schools) treatment of a topological proof (or an almost-proof, up to some intuitively obvious technicalities that should be cleared up in an analysis course) of the unsolvability of the generic quintic in radicals. Unfortunately I remember neither the proof nor the source, and it might be just my imagination...

EDIT: Here is the text (not by Arnold, but based on Arnold's lectures). It is much longer than what I had in my memory, although the price tag of over $100 is questionable... You can get the Russian original for free, but then again with some rudimentary Russian you can just as well get the translated book in djvu... PS. I got from chip-firing to Gröbner bases through a curious and tremendously useful mathematical fact, the Newman lemma (often also called diamond lemma by algebraists, whereas computer scientists use "diamond lemma" for a much easier version of this fact), which is (sometimes) used in proving the basic facts of both of these fields. While it can be avoided in both chip-firing and Gröbner bases, I think it should at least be mentioned (the proof is a wonderful exercise on algorithmic thinking) for the sake of general education. • Here is a link to an English translation of Arnold's book (or at least a substantial part of it): www.nairanalytics.com/abel.pdf – KConrad Dec 23 '11 at 8:54 Combinatorial Nullstellenzatz is a great topic. You can combine it with Dwir's ideas and some other stuff where the key idea is to construct an impossible polynomial though there is no mention of any polynomial in the original problem setup. Let me know if you want more details (I've got to run now) :). You can do Monsky's theorem, that a square cannot be divided into an odd number of equal area triangles. On the way you will have to do 1. p-adic numbers, 2. Sperner's lemma, 3. present$\mathbb{R}$as a vector space over$\mathbb{Q}\$.

In case if you have more time, you could

• use Sperner's lemma to prove Brouwer's fixed point theorem.
• use (3) to do Dehn Invariants
• and I am sure you can find what to do with p-adic numbers

2 by 2 Markov chains? (Don't formally define eigenvectors etc at start; just introduce the idea of the matrix as an update rule for some kind of "dynamical system", get them to do some calculations and make some guesses, then do some ad hoc proof of convergence to equilibrium in non-degenerate case.)

It really depends on what is the objective of the summer camp, and what kind of students are recruited. Are the students selected based on their competency in mathematics, and the goal is to convince them to pursue mathematics in college? In that case, it doesn't help to have a course of more of the same, like a course in geometry, conic sections, or even set theory, as has been suggested. That would only reinforce the view in these students' minds that all the great math has already been done centuries ago.

I see a lot of great suggestions here already, and I would add a recommendation of exploring the mathematics of origami. There have been many recent discoveries in that field. The obvious advantage is that it deals with the obviously beautiful.

I have successfully taught a course for gifted high school students (somewhat shorter than yours, about 9 hours) devoted to the probabilistic method (based, naturally, on Alon and Spencer + some other material). I managed to cover the basics, second moment method, some random graphs, games and derandomization. With a little more time I would have squeezed in the Lovasz Local Lemma.

There was a lot of problem solving, but I was also able to show them some more advanced techniques. In general, combinatorics seems to be a good context to introduce some nontrivial probabilistic tools (say, Chernoff-type bounds).

Another probability-based course in the similar format was "random walks and electrical networks". Very nice topic, quite elementary^1, lots of physical intuition - and at the same time, points at the more advanced math beneath (Markov chains, spectral graph theory)

1 - until the kids ask you "wait, how is this probability on the set of infinite trajectories defined? ;) Luckily, I managed to avoid invoking the Kolmogorov extension theorem.

Some elementary graph theory with the intent of solving traversal or traveling salesman type problems is pretty easy at that level. Introducing incidence matrices can also be a foothold for learning matrix multiplication.

In similar situation I gave courses:

Groups and combinatorics (Polya theorem}

Semigroups and automata

When I was at that stage, I really enjoyed some introductory lectures in set theory a la Cantor in two weeks you can probably get to Schroeder-Bernstein or thereabouts...