We were packing for a trip recently. I have developed a system for getting the kids packed. It is beautiful. Here’s how it works:

- Send kids to basement to get suitcases.
- Keep suitcases on first floor.
- Send kids upstairs to get one type of item at a time. E.g. Three pairs of underpants. Then three pairs of socks. Et cetera.
- Kids throw each type of item in the suitcase.
- Repeat steps 3 and 4 as often as necessary.
- Done.

Seriously. It’s awesome.

I made an observation with Tabitha partway through.

Me:Isn’t it strange how apairof socks is two socks, but apairof underpants is only one thing?

Tabitha(six years old): Yeah. It should “a pair plus one” because there are three holes.

Me:Wow. I hadn’t thought of that. So how many holes does a shirt have?

T: Three….No four!

Me:How do you figure?

T: The one you put your head through, the arms, and the head hole.

If you are like me, you may be a bit behind the curve on her language here. “The one you put your head through” is the one that ends up at your waist once your shirt is on. I had to think about this for a moment.

A few days later, I was curious to probe her thinking a bit further. She was getting dressed (a process which is always slow, and occasionally very frustrating for the parents):

Me:Do you remember how you said a pair of underpants has three holes and a shirt has four?

T: Ha! Yeah!

Me:I was thinking about that and wondering whether there are any kinds of clothing that have one hole or two holes.

T: Socks have one hole!

Me:Oh. Nice. Sometimes Daddy’s socks have two holes, though.

T: Yeah. When they’re broken.

By this time, she finally has the underpants on and her pants are being slowly pulled on.

Me:Wait.Youneed socks!

She goes to her dresser and proceeds to sort through the very messy sock drawer.

T: There are no matches.

I find what appears to be two socks balled up together.

T: No! Those aren’t socks! Those are for putting over tights to keep your legs warm.

We look at each other.

Big smile.

T:Thosehave two holes!

# So what do we learn?

When I posted this on my math education blog, some of my mathematician readers quibbled with me over the number of holes Tabitha and I counted is consistent with how they are counted in higher mathematics. THIS DOES NOT MATTER!

In this conversation, Tabitha and I are considering properties of the things in front of us. Then we are identifying a property and looking for something that has it. We are imagining mathematical properties of things we cannot see. We are playing with ideas.

The formal topology definition of a “hole” is not important here.

So DO NOT let the idea of being wrong get in the way of your math conversations. DO NOT be afraid to play around with ideas you know little about.

Just notice and play together.

# Starting the conversation

This conversation really only took off when Tabitha noticed something that I had not. She noticed that underpants have three holes.

When your child notices a number of something, go with it. Ask follow up questions about other things that are like that, and about things that are different. Here I asked, *How many holes does a shirt have?* and then later, *Are there things that have only one hole?*

Ask questions when they notice numbers in the world, and listen to their ideas when they answer.

How many holes does a hat have? I would say one. Topologists would say none?

Would you get the same count of holes for underpants if you asked your son?

Broken socks is great!

Do winter gloves have only one hole (the whole hand goes in) or do they have six? I’d be curious about Tabitha’s opinion on this.

Did you ever find out about the glove holes?

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I guess one could translate the informal ‘holes’ to the formal notion of ‘punctures’: those leg-warmers are twice punctured spheres. Likewise, a t-shirt is a sphere with four punctures.

Nice 🙂