It’s Sunday morning. Summer has arrived. We are enjoying a beautiful morning on the front porch. I am finishing my coffee. Tabitha (4) has finished her donut.
Then she asks,
Why don’t circles have tips?
What do you mean?
Why don’t circles have tips?
What do you mean by tips? What shapes do have tips?
Triangles and stars. Why don’t circles have tips?
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.
But what if a circle did have a tip?
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.
[Thinks about Math 1060 and the van Hiele model and the mathematical development of children]
Do you want the real answer?
Yes.
OK. Here’s the real answer. See this plate?
It’s circular. Its edge is a circle, right?
Some plates are shaped like a fishy.
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.
What if there were spines?
What do you mean?
What if there were spines all around the circle?
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.
What if they were all around the circle?
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?
Yes.
What about carousels? They are circles and they have points.
I don’t understand what you mean.
What about carousels? They are circles. They have horses on them; those are like tips.
Oh. Right. The circle is just the edge of the carousel. The horses aren’t part of the circle.
Oh.
What got you thinking about circles, anyway?
[points out the window]
What are you pointing at?
[smiles]
I don’t get it.
The tree!
What about the tree?
The bark!
I don’t get it. What about the bark made you think about circles?
It looks like a circle.
Do you mean if you cut the trunk, the bark around the edge would look like a circle?
Yes.
And that circle would have tips?
Yes.
Tabitha starts 1/2-day Pre-K in the fall. She’s going to be a handful.
Overthinking my teaching 
Loved reading that conversation. Tabitha is a serious thinker and will be the most delightful kind of handful possible! Mia has her geometry home show today. She really wishes you could be there. She has been working on a geometry project for the last couple months where she had to design a house, “order” all the materials for the house, keep a budget and build a model of the house. There were many other steps involved. Anyway, today is the home show where they try to sell the house. I have strict instructions NOT to buy her house, but am invited to peruse!
She could come and teach Math 1060. I feel a little sad that I had the same questions for you that your 4 year old daughter did!!
Not at all, Angela. That’s what Math 1060 is for. I almost sent you an email the other day to make sure you’d see this post. I knew you would dig it. Glad you found it on your own.l
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Oh, baby! I hope my kids (offspring) ask these kind of questions.
Actually, I hope my kids (students) do, too! Love it!
Yeah, Mr. V, these conversations are all kinds of fun! I’m gonna argue that the key to having them is listening and taking each question seriously when it arrives. If they know you’re going to give a real answer, they’ll keep asking.
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