I have eight-year-old twins—a boy, L, and a girl, M. One of my many hopes for them is that they grow up to *enjoy* math. They don’t have to ace their math classes, or study math in college. If they enjoy it, I will consider my work (in that department) done.

I think we’re on track so far, but the road has been a little bumpier with M. (I won’t go into too much detail here, but I have a theory—by no means original, I’m sure—as to why boys have historically been perceived as being “better at” math than girls.) I did have a wonderful exchange with her last night, though, and wanted to share that.

She has asked me many times if I would come into the bathroom while she’s in the shower so she and I can talk. I usually demur, because of both the weirdness/awkwardness and the likelihood that our conversation will distract her from actually washing herself. But last night she insisted that I comply because 1) she wanted to talk about *math*, and 2) while I’m on sabbatical she wanted to get at least a little one-on-one daddy-daughter time every day.

It’s like she knows exactly what buttons to push to get what she wants. (I don’t mean to imply that she was being insincere, but I do sometimes notice how easily I can be manipulated.)

Once she was in the shower, she called for me.

“So. What do you want to talk about?”

“Well, you’re the math teacher, so…”

“You want me to ask you a question?”

“Yes.”

“Okay… Well, here’s one that I’ve asked you about in the past, but I don’t think we ever came up with a solid answer.”

“Okay…”

“Is the sum of two even numbers always an even number?”

“… Yes.”

“Why?”

“Becaaause… 2+2 is 4, 4+4 is 8, 8+8 is 16, 16+16 is—”

“Okay… do the two numbers you’re adding have to be the same?”

“Oh. No. Because, like, 2+4 is 6. Which is even.”

“Okay. Well, so, it’s true in all those cases, but that doesn’t mean it’s true in *all* cases. Maybe it’s only true for numbers less than 100.”

“Okay, well, *you* come up with numbers where it doesn’t work.”

This was my first favorite moment. She put the onus on *me* to find a counterexample! I then taught her the word “counterexample,” and said, “I can’t think of a counterexample. But that doesn’t mean there *isn’t* one.”

After a little more thinking, she took a stab at an explanation. It started with, “Well, if two even numbers *didn’t* add up to an even number, then they would add up to an *odd* number. And odd numbers… are odd.”

“Okay… so what does that mean?”

“Well, ‘odd’ means ‘weird,’ so…” Even though I couldn’t see her face, I could tell she was smiling. She suggested that we table that question for the moment. So we did.

This was my second favorite moment. She was perfectly fine leaving the question unanswered. There was no, “Just TELL me!!!” or any kind of frustration. She just set it aside, perhaps to be picked up some other time.

“Do you want a shape question?”

“Yes! I love shapes! What kind of shape?”

“Well, what kind of shapes do you like?”

“Um… I like circles best.”

“Okay… How about this. Picture a square. Then picture a circle inside it that’s touching all four sides.”

“Okay.”

I proceeded very slowly. “How much of the square’s area does the circle take up?” (We hadn’t talked much about “area” before; I was relying on her more “natural” understanding of the word.)

“… Are you counting the *inside* of the circle?”

Favorite moment #3. It makes perfect sense to me *now* that she would think of a circle as *only* consisting of the drawn boundary, but it took me a few seconds to catch on, since in my head, of *course* we were talking about the inside. It’s an *area* question! So that was a cool glimpse into her perception of shapes.

“Yes.”

“… I… don’t even know how to *answer* a question like that!”

I assumed she meant a question that asks how much of one thing another thing takes up. “You could use percents.” (For whatever reason, percents are the part/whole concept they are currently most familiar with.)

“Okay… um… I think 60%.”

“How did you get that?”

“I think each of the little triangle thingies in the corners is 10%, so all together they’re 40%, so… 60%.”

“Ah. Interesting.”

“Actually… the corner things might be more like 5%.”

“Oh. Well, what would your answer be then?”

“… 80%.”

“Mmm. So you think the answer is somewhere between 60% and 80%?”

“Yeah. But… closer to 80%, I think. Like maybe 75%.”

“All right. So… how could we figure out how close we are to the right answer?”

Here she did a dramatic flourish with the curtain, and poked her head out. “It’s… *not possible*.”

“Really?”

“Yeah. Maybe we could invent a machine that takes a drawing you made, and… makes a copy… like it would be thick paper, like almost like cardboard, and it would put lights in each shape, and it would talk to you, and say, ‘*This* shape is 30%.’ Maybe we could email that to some engineers.”

“Yeah, maybe.”

Awesome.

“And then we could do… like, what about a *square* inside a *circle*?”

“… Oooh! That is an awesome question.”

Favorite moment #4. Asking “what-if” questions is something I try so hard to get my *high school* students—even the most “advanced” ones!—to do, and it’s like pulling teeth!

“So what do you thi—”

“I think it’d be the same percent.”

“Huh. Okay.”

At this point I checked in with her about her shower progress, and she confessed that she had not been doing any shampooing or soaping that whole time, so I had to leave.

As awkward—and probably untenable—as the shower context is, one thing I really like about it is that our conversations are necessarily entirely verbal. She can’t reach for pencil and paper (not that there’s anything wrong with that), and can’t even show me anything she’s doing on her fingers. That forces a kind of thinking she might not get to do too often.

Anyway, “Math Chats with M” might wind up being a category or tag on this blog. So far we both enjoy them, but I think we’ll have to be more intentional than ad-hoc about the venue.

Lovely. Now I want a view into the alternate universe where you said “No” to counting the inside of the circle.

Oooh. Me too! I’m sure we’ll return to the question (or one like it) before too long…