I had set theory in third grade (involving cows, pigs,etc.) and I loved it, but it never got integrated into the rest of the curriculum and I can't say I got a lot of use out of it.

However, I don't remember really loving set theory or boolean algebra.

There were many things that I really enjoyed in math class, such as working with different bases (1st or 2nd grade), algebra (7th grade), geometry proofs and trig (8th grade), statistics and the first few units on differentiation and integration (high school), and linear algebra and its uses in image processing and structural analysis (college).

I think the difference between topics that were enjoyable and felt mind-expanding and made me want to be a mathematician, versus topics that were intimidating or just tedious and made me wish I were doing something else... was the teaching. A good math teacher can take almost any topic in math and make it about DISCOVERY. Before even revealing what is being taught, the teacher states a problem that can be solved using the upcoming material. You FEEL that there is a unique answer, but you don't know how to get to it, and now you WANT to learn how. If a specific and simpler form of the problem is presented, maybe the students can solve it using common sense, and then the teacher can ask "How do things change when we add this complexity? How could we deal with it?". A technique is proposed either by the teacher or by students, and is used successfully. The teacher then asks "Will this work every time? Can anyone think of a time when this would not work?". Someone - teacher or student - comes up with an example where that technique wouldn't work. Then they either try to patch the problem-solving methods further, or say "Solving that kind of problem would require much more complicated techniques. They're called such-and-such if anyone wants to look them up later, but we'll only teach this in the classroom about a year from now". Everything from division and fractions (2nd grade) to multivariable calculus can, and should, be taught this way. THAT is what thinking like a mathematician is like.

I didn't enjoy boolean algebra or set theory because they were taught by bad teachers who went "Now we are going to learn X. Here are some definitions. Here are some properties between these kinds of things. So for example, when a problem looks like this, you get to the solution by doing this. Got it? Now do five examples by yourself and raise your hand if you have a question". Bo-ring!

I loved graph theory, though, which I did in 4th grade. The fact that all polyhedra we could think of (except ones with holes) have an Euler Characteristic of 2 seemed like MAGIC to me. Why was this true? The proof involved flattening the polyhedra onto a plane. The equivalency between solids and these 2D diagrams was super interesting. And then the proof (removing either one edge, or X edges and X-1 vertices, in either case combining two faces into one, until you have two faces - an "outside" one and an "inside" one bounded by X edges and X vertices) was AMAZING. It was the first time I had been exposed to a proof, and to the idea that you could pick apart WHY something was true in math. You could OWN the truth of the mathematical process in a way I had never experienced. Because it was a process of discovery, not just a simple fact.

So, yeah, as long as things are taught that way, I think kids will appreciate "mathematician thinking" even without learning topological or computational topics like these. (And by the way, everyone here has read Paul Lockhart's "A Mathematician's lament", right? If not, Google it! It makes the same point buch much more richly).

I'm not sure I agree that teaching chaos theory is totally a good idea... it's certainly a fun thing to have a one-off lesson on, since the ideas are fun and graspable, and there are lots of exciting pictures... but to actually solve any problems in it, and understand what you're doing, is pretty hard.

I nominate Pick's theorem as a great topic for schools. It's genuinely useful, especially in the computer age, is easy to apply with an unexpected but still fairly simple statement, and its a great example of a mathematical proof. I have some slide on it here: http://richardelwes.co.uk/wp-content/uploads/2012/02/PickEhrhart.pdf