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.

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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.

# Postscript

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.