"Mathematics talk" for five year olds

I'm going to quote Bill Thurston from his interview for More Mathematical People:

Thurston: ... One thing that is very important is the education of children... In the elementary schools in Princeton that my kids have attended, there is an annual event called Science Day. They bring in scientists from the community, and we spend a day going around from class to class talking about things. I have enjoyed doing that quite a bit.

MMP: What have you talked about?

Thurston: I have done different things every year for ten years or so; for example, topology, symmetry, binary counting on fingers... I find that kids are really ready to pick up mathematics in the way that I myself think about it. Of course, it's toned down.

MMP: Can you be a little bit more specific about the way you think about mathematics?

Thurston: That's a tough question. It might be nice to give an example. At one time I went into a class of kids and made lots of equilateral triangles. We made a tetrahedron by putting three triangles at each vertex. Then I asked what happens if you put four triangles, and they constructed an octahedron. Then with five triangles at each vertex they constructed an icosahedron. But with six triangles they found that the construction just lays flat. And then I asked about seven triangles at each vertex. They pieced it together and they got these hyperbolic tesselations in four-space. They loved that. The kids did. But the teacher really felt ill at ease. She didn't know what was happening.

I've seen a very successful interactive math talk to a participating audience of five and six year olds at MSRI a few years ago. It was structured around the question of "what's the largest number?" Kids had fun coming up with large numbers but eventually one of the youngsters figured out a simple way to always come up with a number bigger than the previous one named. Through some nice leading by the teacher, they eventually concluded (indeed proved) that there are an infinite number of natural numbers. It was fun watching the kids (by then fully engaged) grapple with what is really a quite abstract idea --- that one can know that the numbers are unending without actually exhibiting them in any way. It was a terrifically sneaky way to engage kids and hear their ideas on notions like "infinity" or even "number".

Giving a little "Magic show" about Möbius strips might be fun. Make a huge number of Möbius strips and cylinders, and hand them out to the kids along with safety scissors. You could have a predrawn "center line". Ask them what will happen when they cut along the center line: How many pieces will you get? They will probably say two for both shapes. Have them cut along the line and see what happens! The result for the cylinder is as expected, but for the Möbius strip you get a piece of paper with two twists. Now ask them what happens if they cut this in half. You get two interconnected links! You could have them start with a new Möbius strip and cut halfway between the center line and one edge, following their way around. You get a Möbius strip linked to a double twisted strip! This will all be great fun for the kids.

You can "explain" some of these phenomena by showing them how to make a Möbius strip themselves: just take a strip of paper, twist it, and tape the ends together. From this perspective cutting a Möbius strip in half is just the same as taking two strips next to each other, twisting both of them, but the head of one piece attaches to the tail of the other, so you can see how cutting the strip in half only leads to "one piece". It might help to have some different colors of paper, so they can more easily keep track of the "two halves".

You could show them some Escher drawings involving Möbius strips - the one with the ants would probably delight them.