My approach to having (usually math-related) conversations has generally been to keep my eyes and ears open and jumping on any opportunities. But not all the time. Here's the story of what happened when I set out something I truly hoped she'd interact with in a mathematical way.
I took them home. Later in the afternoon I washed them. I waited to see if she'd notice them. SHE DID!!
The minute she saw them she was FULL of ideas. A mile a minute, the ideas gushed out of her: "Let's sort them into categories! I'm going to make a zoo! I'm going to make a pile of cats and a pile of dogs. And we'll give all the horses to Alice..."
"...now I'm going to put them into lines. The darker colors here, the lighter colors here. See? Take a picture of the cats, too. But make sure NO other animals are in the picture."
And here I'll pause to say that her inclination to sort, classify, arrange, and organize by attribute seems to me to be the essence of mathematical activity, and a thinking skill useful and applicable at all levels of mathematics. Over the course of the next hour she narrated what she was thinking and what she was going to do next. During this time I had ample opportunity to listen to her reasoning, ask her questions to challenge her thinking and clarify my understanding of the choices she was making as she physically and mentally sorted the animals within the framework of her elaborate narrative.
She had decided to make a zoo. Each category (her words) of animal would be in its own area and assigned a color. Here's where things got interesting. Green was the color for all the cats, but the "tame cats" got a second color as well (purple). Wildcats were in a different pen but still in the green category. The lone mouse got put into the cat pen "so the cats can play with it."
Orange was for giraffes. Elephants were light blue. Dinosaurs "will need a big area because they're, well...dinosaurs. Birds will go with dinosaurs because they're part of the same family."
Her plans were big: "We'll record how many of each and make a graph. I'll put them in arrays of fives if possible." (I love that in addition to the sorting and classifying she also thought of them as potential data and that they could be organized in arrays to help the counting process).
For the most part she stuck with the arrays (except for a few categories). In the picture above she sorted the dinosaurs into lines of small and big, but not the rhinos. When asked about this, she didn't really want to talk about it. Too busy with the next thing, I think.
Then came the sea creatures. She put them in a bucket of water because "they just need a place to swim." She started to put the crocodiles and turtles in too. "Wait a minute!" I said. "Those are amphibians. They live in water and on land. Do they belong with the sea creatures?" She decided she'd sort the amphibians out another way, but decided to add a yellow mark to the sea creatures "so they don't get confused for the amphibians. And, they might wander off."
She had more questions as she sorted:
Do ducks count as amphibians? [Why do you ask that?] Well, they are on land and in water.
Do penguins count as birds?
Are hippos amphibians? [Why?] Cause they go in water too. I always thought they were amphibians.
As she continued she built an extension for the zoo to make room for the 30 horses.
Earlier, as I washed and sorted the animals into loose categories (that's just how my brain rolls) I had wondered what she might do with all these creatures and, if it was sorting, what kinds of categories she might choose. I started my sort by grouping by animal type (zebras, horses, cats, dogs, dinosaurs, etc.) but pretty soon created categories to include a wider definition. So, in addition to cats and dogs I also had "farm animals," "sea creatures," "animals from Australia," etc.
That's how it went for her too, although our categories differed. What I called sea creatures ended up as "water animals" and "amphibians" for her. There was also a fascinating category she initially called "under three" which was a catch all for any small animal group. All creatures with "under three" members would be put in a pen together. I got confused and thought it meant "three and under" but she was clear "under three" meant TWO. I asked her why she hadn't called it "two and under" but I got the brush off.
It took an HOUR of sorting, arranging, categorizing, planning and organizing for her (and about a million mosquito bites for me) before her zoo was complete. I was exhausted. The kid, on the other hand?
"And now!? We need to organize the tours!"
And here I'll make a pitch for Talking Math with Your Kids and Doing Science with Your Kids ('cause you know we did both, right?). Both sites (and conversations via Twitter hastags - #tmwyk and #dswyk) provide incredibly strong, real-life models for how to have mathy and science-y conversations with your kids just by listening to what they have to say and going from there. Don't delay, talk science and math with your kids today!