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.
This is really rich, and of course, there are places I wanted it to go that it didn’t (which is not a problem for me, just a wish for multiple time lines and gardens of forking paths a la Borges). For example, I thought, perhaps wrongly, that she was pushing towards a limit situation when she asked,
“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?”
And since she pushed it with her follow-up, I’m willing to guess that she was envisioning at least the beginning of things that come up when we ask students to investigate n-gons as n approaches infinity. Of course, not with that language, but “ALL around the circle” is different from “a lot of spines around the circle” and she says it twice. Then don’t the spines and their tips converge to the circumference of a circle and meet the definition?
Not sure how I’d have gone with that since I don’t know your daughter and wasn’t there, and it’s impossible to run the tape back and ask new questions or make new comments. But this is one of those Deborah Ball moments for me, when it’s not that something wrong happened, but that other intriguing right things could have happened (though this idea might have gone bust in practice). Better to have many potentially productive choices, for my money, than to have one rigid goal in mind towards which you lead a child or class in lockstep no matter what. And of course, not fatal to pick a road that doesn’t bear fruit you’d hoped for as long as you are prepared to retrace steps to the last major fork.
One question: how are you preserving all these amazing conversations? You’ve probably said more than once and I’ve overlooked the explanation (in which case, just tell me to do my homework). But regardless, this material is pure gold. Then again, it helps to be the person who knows s/he’s going to be talking math with his/her kids. ;^)
For what it’s worth, I was anticipating a conversation about bicycles when she asked about the spines. I was seeing the spokes on a bicycle wheel.
I’m impressed by the patience – from dad and from Tabitha – shown in this conversation. It gives me something to aspire to with my children (aged 10 and 4) and with my students (aged 14 – 19)
I might have brought in other shapes with equal sides: triangles have 3 sides, squares 4, pentagon 5, hexagon 6, octagon 8, etc. The more sides you add, the less pointy the tips are and the more it looks like a circle. My little guy just turned 4 and is very analytically minded. I’m excited to have these more complex mathematical conversations.
Thanks for weighing in, everyone! I love the additional directions readers bring to these conversations. Keep ’em coming.
The ‘what made you think of it?” is a really good question isn’t it, not so much a mathematical question as a narrative one. Reading this great conversation makes me think I should, with my class, ask what made them think things earlier on.
(Tabitha seemed to be getting to the idea of approximations to shapes, something that comes up with fractals, or in fact with any real-world shapes; if you get a microscope they’re all going to be like that bark. There are so many lovely directions this could take. The idea that a “circle” is something we don’t find if we measure carefully enough, so where is there a real circle?)
Funny to have you point that out, Simon—about that what made you think of it? I find myself asking some version of that question frequently around the house, and it often leads to new and interesting places.
Re-reading this a few years later, I think I didn’t quite understand Tabitha’s question. I think she may have been asking about corners or vertices, where I was imagining that she was thinking about spines—like brush bristles or porcupine quills—sticking straight out. And there is lots of interesting analysis to be done in light of this realization.