Sometimes DD initiates the math, like talking about the cars in the garage. Or she likes to point out how many of us are wearing blue that day, or how many girls are in the house, etc. But other times, we initiate it. For instance, the other day DH made a play-doh pizza with 8 slices. He gave DD a slice and asked her how many were left. The thing that got us was that she became super enthralled with this game and he had to do it with her for another 10-15 minutes straight, subtracting and adding pizza slices over and over again. We could see DD was really thinking about it and when she's trying to learn a concept, she will do it over and over again (that doesn't seem normal to me, either).
About the tangrams: she annoys the heck out of me because if I am in the room when she pulls out the tangrams, she acts like she needs help. But if I walk into the room after she's already started, she will do them all by herself -- until she realizes I'm in the room! Grr!

She does that with other puzzles, too -- drives me nuts.
She asks DH to count to 100 when he gives her a bath and she seems to understand what it means to go by 10's and 1's because she asks specifically for him to do it those ways.
I got together with my DB and his DD (same age as my DD) and the difference is astounding; I'm always amazed just from a scientific standpoint how different the girls are from each other. DN would say, "They made big mess!" Meanwhile, my DD is saying, "Come on, Kaitlynn. I'll hold your hand so you won't get lost." (Yes, she likes to mother other kids who I think she finds on a mentally lower level than her.) I just find it fascinating, the differences between those two. Not to say I don't love DN because she's ND; it's just interesting to watch the differences. I cannot imagine DN doing any of the stuff DD does when it comes to reading, operating the computer, playing boardgames, counting.
Okay, I kind of went off on a tangent there that was unrelated to my original question.
I cannot believe that other 3-yr-olds do not understand 1-to-1 counting -- is that really true? When are kids supposed to know how to do that? I read something that they expect kids to count 10 objects after completing kindergarten -- is that accurate?