Non-traditional math concepts for early education I am currently working on source material for a math-related software project with my mother, who has a PhD in Elementary Education and specializes in math education. While she has quite a strong fundamental understanding of the current techniques and topics taught to children at an early grade school level, I am relatively unfamiliar with the subject area. Lately, I've been looking for sources of inspiration and information, such as Conrad Wolfram's TED talk "Teaching kids real math with computers".
Remembering that the Stack Exchange family now has this excellent math site, I figured I would ask this question here:
What are some interesting concepts in math that can be taught to elementary school children, that aren't traditionally taught? This can be because the topics aren't considered suitable, or perhaps because no one's thought to do it yet. For example, at the end of his talk, Wolfram describes a technique to visualize calculus using limits and shapes inscribed in a circle.
(While I realize this question is slightly unusual compared to other questions I see on the front page -- in other words, university-level math questions -- I think it is appropriate given what the FAQ mentions as appropriate topics for this site.)
 A: I've had success teaching an elementary school student about binary numbers.  I listed powers of two up to 512 and claimed I could make any number up to 1000 by adding the powers of two, using each one at most once.  I had the  student give me numbers and demonstrated how to make them out of the powers of two. 
Then I had the student do it herself for some small numbers using guess-and-check.  Then we made a table with powers of two as columns and sequential integers, from 0, as rows, and put an X in any entry if the power of two for that column was used to make the number in that row.
We looked for patterns in the X's, and were introduced to binary numbers that way.  By the end of two hours, she could convert any number between binary and base ten, and could add and multiply in binary.
I haven't tried to do it and don't know how, but I suspect you can introduce basic group theory to students of this age, too.
A: Prologue
I am $13$ years old, which is neither advanced nor elementary.  In this post, I'd recall the mathematical concepts that fascinated me a lot in the past years.

A good thing to learn about is elementary number theory. I learnt how odd numbers are in the form $2k  +1$ and even numbers are in the form $2k$, where $k$ is another integer. It was really interesting to tackle the proofs related to this. The realization came to me that remainder is not just another number when I studied modular arithmetic. It was a whole new experience. One is never taught elementary number theory throughout the elementary school (and mostly high school) years! 
Besides that, I learnt about number bases and how to convert them. I also watched the TED talk the OP shared succeeded by buying Wolfram Mathematica. It's a great  software and I use it for manipulations I couldn't do by hand. Also, I used my resources a lot: MIT OCW, Khan Academy, OpenStudy et al.
Teaching an elementary MathJax is enough for him to go to math forums and ask (So did I!). 
A little remark from my side: It'd be really, really good to have a site like Math StackExchange for juniors, where a little leniency is  shown to newbies.

Epilogue
The above post is self-centered (sorry!), but the things I mentioned are good enough for any other elementary.
A: The following is a list of mathematical ideas that I successfully explained to kids (10-12 or so). I list the topics using the formal terminology but, of course, when explained to young kids I usually employed a more intuitive, vivid language and setting. Some of the following took several hours over some sessions to explain. Other are a lot easier. 


*

*Proving that there are infinitely many prime numbers. The standard proof (basically, multiply a bunch of primes, add one, look for a prime divisor, viola: a new one) requires very little facts about divisibility that are easily explained. The entire logical sequence of the proof is rather short. A preliminary introduction to primes, some examples, and then the question: "how many are there" are usually quite well appreciated. 

*Proving that the cardinality of the even numbers is the same as all naturals. A preliminary discussion of how to compare finite sets without counting easily leads to the notion of bijection as the means for comparison. Then, listing the evens and the naturals in a bijective form is very illustrative and easily digested.  

*Proving that a countable union of countable sets is countable. This may be quite a bit trickier but doable. Starting off with union of two countable sets is countable is a good idea. Of course, the whole discussion can be placed at the Hilbert Hotel making things more vivid. 

*Proving that $[0,1]$ is uncountable. The way I usually approach this is, instead of using the real numbers, to introduce into the Hilbert Hotel a kid into each room. Each kid has an infinite countable list of his/her toys (where there are only two kinds of toys in the world). Then showing that it is impossible that all kids with all possible lists of toys reside in the hotel using the standard diagonalisation. 

*Proving that $\sqrt 2$ is irrational. I managed to do that successfully only one with a young kid so maybe it's too advanced. 

*Proving Sperner's Lemma. Here is the proof I use: Draw a triangle with a triangulation of it. Draw numbers on the vertices according to the statement of the lemma. Treat each small triangle as a room, with the edges as doors with locks of type 1-2, 1-3, or 2-3 according to the numbers on the vertices. Declare that any room numbered 1,2,3 has a great party going on in it. The kid is looking for the party and is given a 1-2 key. Convince the kid that a way into the complex exists along the long 1-2 edge. Use the key to go in, and explain that a door seals itself forever after you go through it. Then either you found a party or you can get out of the room. Then either you are out of the complex, in which case you can go back inside, or you found the party. Keep going until party is found. 

*Pythagoras' Theorem (with at least 3-4 different proofs). There are so many accessible proofs I won't say more. 

*Exploring some ideas of spherical geometry (no parallel lines, exploring sums of angles of triangles etc.). I once sat down with a young girl and saw she got really bored with the school math I was helping her with. So, I took an orange and carved some straight lines onto it and started asking her questions about it. Before long she was going through the entire bowl of apples and oranges carving all sorts of things on them. 

*Constructing an approximation of the hyperbolic plane by cutting several identical paper annuli, chopping them down into reasonable pieces and gluing them together allowing the curvature to take effect. Then, explore lines on the model. Easy and fun. 

*Playing with a Moebius streep: at least cutting it along the middle and then cutting it along 1/3. Easy and fun. 

*Playing with identification spaces. Listing all those obtainable from a square, thus including the Klein bottle. Easy and fun, especially if you explain that the identifications are actually done by using tele-porters. 
A: I'm a fan of Group Theory. Like, you go to school, and they teach you all these rules for how to add and subtract numbers. And then Group Theory comes along, and you can just make up a bunch of rules of your own!
But then, if you want the resulting system to have interesting properties, your made-up system has properties like associativity or commutativity, and as you add these, suddenly familiar rules from normal arithmetic reappear seemingly out of nowhere.
I like the idea of showing people that mathematics isn't about just memorising rules; you can make up new rules and follow them to see what the consequences turn out to be. (But it's a bit like designing a game; most choices of possible rules turn out to be rather boring, until you figure out how to come up with interesting choices.)
I suspect you could probably draw a Cayley table and get small children to understand it. You could probably also explain the dihedral groups or the permutation groups without too much trouble. All you need is a clock face to demonstrate mod 12 arithmetic, and how some numbers are divisible and some aren't, and so on.
That said, I can imagine some children being utterly baffled about what the heck you're doing with all these weird new symbols, and how you seem to be just pushing them around on the board at random. And the concept of what a "group" actually is... that's really, really abstract. I wouldn't expect young children to grasp that. So it's not like you'll be describing cosets or solvable groups or anything.

If you know that multiplication is repeated addition, you can probably grok that exponents are repeated multiplication. And you can maybe wrap your mind around logarithms. Deriving the first and second laws of logarithms, and their consequences (e.g., what is $x\uparrow \frac12$?) is an instructive exercise in mathematical deduction.

Last suggestion: I wasted many hours playing with WinAmp, particularly the programmable visualisation plugin that comes with it. [Note that WinAmp has apparently been discontinued. You can probably still download it somewhere.] By throwing formulas into it, you can make it draw trippy stuff. Again, not sure how well really young children would grasp what the heck is going on here.
