Yesterday I showed this picture to my kids and asked whether they thought there were more black ants or more red ants. Both of them immediately wanted to start counting the ants, but I stopped them because I was interested in how they thought about it *without* counting.

My kindergarten-age daughter initially thought there were more red ants, because the arrangement looked “longer”, and indeed when she counted how many ants there were in a row, she found that the rows of red ants were longer and was satisfied that there were, indeed, more red ants. But when (with a little prompting) she counted the columns, she seemed surprised to count a longer line of *black* ants. She seemed more delighted than puzzled at this outcome, and was perfectly content to leave the question unresolved.

My 2nd-grade son decided pretty quickly that there were the same amount. He had seen his sister count the rows and columns, and in response to the puzzle of the different outcomes, he said, “You have to figure out how many are inside.” He then proceeded to count all the ants around the perimeter, and found 24 in both cases. “So since there’s the same amount all around,” he told me, “you know there will be the same amount inside.”

And then it was bedtime, and so I left it at that, with my daughter not knowing the answer and feeling interested but not overly invested in the question, and my son feeling quite confident in his incorrect, but very reasonable, solution.

The fact that the conversation was interrupted gave me more time than usual to think about what happens next. Do I…

- Have them count the number of ants to see that there are actually 49 black ants and 48 red ants?
- Explore similar problems, with different numbers of ants?
- Manipulate the original pictures to help them visually see the difference between black ants and red ants?
- See if
*they*can create configurations of ants that push up against their initial theories?

When confronted with a “how would you respond?” question, my college students often ask, “But what is the *right* way to respond?” or “What is the *best* way to respond?” This is a hard question to answer! There are often some good ways and some not so good ways that you could respond (and sometimes a “good” way can backfire, and occasionally a “not so good” way can turn up something surprising). I don’t know that you can know the *right* way to respond, and sometimes you’ll experiment and it won’t go so well and then there’s always a next time.

I haven’t decided yet what direction to take (I’ve gotten lots of great ideas from folks on Twitter), but while whatever I do may or may not be “right”, I do have some general principles. I want to work with the thinking they’ve already given me rather than impose my own thinking on them. I want to spark their curiosity. I want to be flexible enough to go a different direction if that’s where their interest takes them. And I want whatever I do to happen in an environment of love and care.