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.)
Best Answer
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.