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:

  1. Send kids to basement to get suitcases.
  2. Keep suitcases on first floor.
  3. 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.
  4. Kids throw each type of item in the suitcase.
  5. Repeat steps 3 and 4 as often as necessary.
  6. Done.

Seriously. It’s awesome.

I made an observation with Tabitha partway through.

Me: Isn’t it strange how a pair of socks is two socks, but a pair of 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. You need 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.

TThose have 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.

A circular conversation

The following conversation took place about two years ago. It is probably the first one that made me realize how important it is to talk math with my kids. Near the beginning of the conversation I noticed myself making a choice between engaging her mind and moving on to other things.

That choice—and the knowledge needed to notice it, and to follow up on it—has become interesting for me. Through this website, I hope to share what I have learned about that, and to learn more through interaction with readers. So please send reports of your conversations to me. And get those questions to me, too. You can do both through the About/Contact page.

It’s Sunday morning. Summer has arrived. We are enjoying a beautiful morning on the front porch. I am finishing my coffee. Tabitha (four years old at the time) has finished her donut.

Then she asks,

Tabitha: [four years old] Why don’t circles have tips?

Me: What do you mean?

T: Why don’t circles have tips?

Me: What do you mean by tips? What shapes do have tips?

T: Triangles and stars. Why don’t circles have tips?

Me: Well…that’s a good question. I guess that’s part of what makes them circles. If they had tips, they wouldn’t be circles.

T: But what if a circle did have a tip?

Me: Well, then it wouldn’t be a circle. I guess what makes a circle is that it’s round. If it had a tip it wouldn’t be round.

There is a pause, during which I realize that I have not really given Tabitha my all with that explanation.

Me: Do you want the real answer?

T: Yes.

Me: OK. Here’s the real answer. See this plate?

It’s circular. Its edge is a circle, right?

T: Some plates are shaped like a fishy.

Me: Right. Good.

But this one’s circular. There’s a point in the middle of the plate; that’s called the center. All the parts of the plate on the edge are the same distance from the center. If there were a tip, then the part at the end of the tip would be farther from the center than the other parts, so it couldn’t be a circle. What really makes a circle a circle is having all parts be the same distance from the center.

T: What if there were spines?

Me: What do you mean?

T: What if there were spines all around the circle?

Me: Well then the tips of the spines would be further from the center than the base of the spines, so it wouldn’t be a circle.

T: What if they were all around the circle?

Me: Still, there would be parts at the end and parts at the base.

Did you like getting the real answer? That answer about circles being round, that wasn’t really the real answer. Did you like the real one?

T: Yes.

There is a thoughtful pause.

T: What about carousels? They are circles and they have points.

Me: I don’t understand what you mean.

T: What about carousels? They are circles. They have horses on them; those are like tips.

Me: Oh. Right. The circle is just the edge of the carousel. The horses aren’t part of the circle.

T: Oh.

Me: What got you thinking about circles, anyway?

T: [points out the window]

Me: What are you pointing at?

T: [smiles]

Me: I don’t get it.

T: The tree!

Me: What about the tree?

T: The bark!

Me: I don’t get it. What about the bark made you think about circles?

T: It looks like a circle.

Me: Do you mean if you cut the trunk, the bark around the edge would look like a circle?

T: Yes.

Me: And that circle would have tips?

T: Yes.

So what do we learn?

There is a lot in this conversation. As is often the case, when the conversation began I had absolutely no idea what she was talking about. What in the world could she mean by “why don’t circles have tips?” I work each semester with college students planning to be elementary teachers. I preach to them the importance of patient listening and asking questions to better understand what their students are telling them.

This is a message I frequently need to take to heart.

Tabitha’s questions are about making a transition from what shapes look like to what makes them what they are. She seems to want to know what makes a circle a circle.

This takes place as she thinks about the cross section of the tree in our front yard.

She knows that this would look circular, but that it isn’t a circle. She identifies a property that the tree cross-section has that a circle does not-tips, or sharp points.

I started with a crummy answer. I basically told her that Circles don’t have tips because if they did they wouldn’t be circles. And I felt guilty right away.

So I offered her a real explanation. That explanation was based on the definition of a circle, which is The set of all points a fixed distance from a common point, called the center.

This explanation was one that the average parent may not have ready at hand, though. So what do you do if you don’t know why a circle has no tips (or whether a square counts as a rectangle, or whether it’s still a right triangle if it points to the left, or…)? You model good information-seeking skills. Try to agree on what the question is (What do you mean by tips? What shapes do have tips?) Then consult books and friends and neighbors. You must know someone who has taken high school geometry more recently than you have. Maybe you have an engineer in the family, or a math teacher up the block. Your library has a librarian. Any of these people would be delighted to help out a young child with a geometry question.

And now that you’re reading this blog? You’ve got a friend ready to help. Shoot a note through the About/Contact page; we’ll get you an answer ASAP.

Starting the conversation

This conversation was Tabitha’s idea. The only thing I did here was listen and try to understand her questions.

We can all do that.


Take the time to read the comments. Other parents weigh in with some lovely ideas for additional directions one could take this conversation. The key is that there is not one right conversation to have with your kids. The key is to have that conversation by asking and listening.

The unit is the thing that you count

Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out of the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

So What Do We Learn?

What makes this more than just a funny story is that Griffin and Tabitha are clearly counting different things. They are talking about different units.

When we make cookies, everyone agrees on the unit; we know what one cookie is.

But brownies are different. Tabitha seems to think that a brownie is the thing that comes out of the oven. Griffin seems to think that a brownie is what you eat in one serving.

One brownie according to Tabitha.

One brownie, according to Griffin

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.

Fun follow-up question: Does the thing in this video count as one brownie?

Starting the Conversation

Anytime there are things in groups—or things being cut—is a good time to talk about units.

Grocery stores usually have express lanes where you have to have Ten items or fewerAsk your child whether someone with a dozen eggs could use that lane. What about someone with 12 apples in a bag? What if the apples are loose?

When your child asks for two slices of pizza, take one slice, cut it down the middle, smile wryly and ask whether that’s OK.

In all of these cases, the central question is What counts as one? Play with that question.

Also, watch that video together. It’s a ton of fun.