T was a third grader when all this went down.

At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that $4\times 6$ is four sixes, and because that’s 24, she also knew that six fours would be 24. I asked why she said so and she didn’t know why. I asked her if she thought this would always be true for bigger numbers, or could it be possible that there were some big numbers like 30,001 and 5,775 for which 30,001 5775’s was different than 5,775 30,001’s. She wasn’t sure. I asked her if she thought it was a good question and she said she thought it was.

So this session I reminded her of this conversation. I forget the details of how we got going on it; I remember inviting her to wonder about the question and note that there is something surprising about the equality between four sixes and six fours. She could count up to 24 by 4’s and by 6’s and mostly you hit different numbers on the way up, so why do the answers match? And would it be true for any pair of numbers? But the place where I really remember the conversation is when we started to get into the nuts and bolts:

Me: Maybe to help study it we should try to visualize it. Can you draw a picture of four sixes?

T draws this –

Me: Cool! Okay I have a very interesting question for you. You know how many dots are here –

T: 24 –

Me: and you also told me that six 4’s is also 24, right?

T: yes.

Me: – so that means that there must be six 4’s in this picture! Can you find them?

T: I don’t understand.

Me [writing it down as well as saying it this time]: You drew a picture of four 6’s here, yes?

T: Yes.

Me: And that’s 24 dots, yes?

T: Yes.

Me: And you told me before that 24 is also six 4’s, yes?

T: Yes.

Me: So it must be that right here in this picture there are six 4’s!

[It clicks.] T: Yes!

Me: See if you can find them!

At this point, I go wash my hands. An essential tool that has developed in my tutoring practice is to give the student the social space to feel not-watched while they work on something requiring a little creativity or mental looseness, or just anything where the student needs to relax and sink into the problem or question. The feeling of being watched, even by a benevolent helper adult, is inhibitive for generating thoughts. Trips to wash hands or to the bathroom are a great excuse, and I can come back and watch for a minute before I make a decision about whether to alert the student to my return. I also often just look out the window and pretend to be lost in thought. Anyway, on this particular occasion, when I come back, T has drawn this:

T: I found them, but it’s not… It doesn’t…

I am interrupting because I have to make sure you notice how rad she’s being. The child has a sense of mathematical aesthetics! The partition into six 4’s is uglifying a pretty picture; breaking up the symmetry it had before. It’s a kind of a truth, but she isn’t satisfied with it. She senses that there is a more elegant and more revealing truth out there.

This sense of taste is the device that allows the lesson to move forward without me doing the work for her. Her displeasure with this picture is like a wall we can pivot off of to get somewhere awesome. Watch:

Me: I totally know what you mean. It’s there but it doesn’t feel quite natural. The picture doesn’t really want to show the six 4’s.

T: Yeah.

Me: You know what though. You had a lot of choice in how you drew the four 6’s at the beginning. You chose to do it this way, with the two rows of three plus two rows of three and like that. Maybe you could make some other choice of how to draw four 6’s that would also show the six 4’s more clearly? What do you think? You wanna try to find something like that?

T: Yes! [She is totally in.]

At this point I go to the bathroom. I hang out in the hall for a bit when I get back because she seems to still be drawing. Finally,

Me: Did you find out anything?

T: I drew it a lot of different ways, but none of them show me the six 4’s…

She’s got six or seven pictures. One of them is this –

Me: Hey wait I think I can see it in this one! (T: Really??) But I can’t tell because I think you might be missing one but I’m not sure because I can’t see if they are all the same.

T immediately starts redrawing the picture, putting one x in each column, carefully lined up horizontally, and then a second x in each column. As she starts to put a third x in the first column, like this,

she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:

The pieces just fell into place from there. Again I don’t remember the details, but I do remember I asked her what would happen with much bigger numbers – might 30,001 5,775’s and 5,775 30,001’s come out different? And she was able to say no, and why not. Commutativity of multiplication QED, snitches!