Talking Shapes with Kids

I have been spending time talking with kindergarteners, first and second graders in schools about my shapes book (coming from Stenhouse, Spring 2016). Many more school visits are ahead of me. I have written up some reflections for a more teacher-ish audience than this blog attracts. If you’re interested in the ways young children talk about shapes, and in what I hear in their ideas, hop on over to the sister-blog Overthinking My Teaching for the details.

You may be delighted to learn how much math there is in the simple collection of shapes below.

Building a better shapes book [Which One Doesn’t Belong?]

IMPORTANT NOTE: The moment alluded to below has arrived! Which One Doesn’t Belong? is now available from Stenhouse as a student book (awesome for home reading, too!) and a teacher guide.

As a result, I have removed all links to the version I was previously distributing free.

There are many shapes books available for reading with children. Most of them are very bad. I have complained about this for years. Now I have done something about it. Most shapes books—whether board books for babies and toddlers, or more sophisticated books for school-aged children—are full of misinformation and missed opportunities. As an example, there is nearly always one page for squares and a separate one for rectangles. There is almost never a square on the rectangles page. That’s a missed opportunity. Often, the text says that a rectangle has two short sides and two long sides. That’s misinformation. A square is a special rectangle, just as a child is a special person. After years of contemplation, I had a kernel of an idea the other night. The kids are back in school before I am, so I had some flex time available. One thing led to another and voilá. A better shapes book. (Links removed—see above note.)

How to use this book

On every page are four shapes. The question is the same throughout the book—which one doesn’t belong? For example, which shape doesn’t belong in this set? If you are thinking, “It depends on how you look at it,” then you’ve got the idea.

• The bottom left shape doesn’t belong because it’s not shaded in.
• The top left shape doesn’t belong because it only has three sides, while the others have four.
• The top right doesn’t belong because it is the only square.
• The bottom right doesn’t belong because it’s the only one resting on a side.

Maybe you have different reasons for some of these. That’s great! The only measure of being right is whether your reason is true. With an infant, you can use this book like any other shapes book. Look at each page together. Point at each shape and talk about it as you snuggle. With a young child, ask which one doesn’t belong and why. Most pages in the book have at least one shape that a young child can identify as not belonging. Join the conversation by pointing out a different shape that doesn’t belong for some other reason. With an older child, challenge yourselves to find a reason for each of the 44 shapes in the book. There is no answer key. This is intentional–to encourage further discussion, and to encourage you to return to the book to try again. I have tested the file out on the Kindle app on my iPad, and it looks good. I made one printed copy and prefer it to the e-version because I can leave it out for browsing and we can touch the shapes without accidentally turning the page.

The legal details

I owe thanks to Terry Wyberg at the University of Minnesota, who regularly plays the “Which one doesn’t belong?” game with numbers in professional development sessions; to Megan Franke at the University of California, Los Angeles, who adapted the old Sesame Street game “One of these things is not like the others?” and to my online colleagues including but limited to Justin Lanier, Megan Schmidt, Dave Peterson, Matt Enlow and Andy Rundquist.

The following Which One Doesn’t Belong? prompts are yours to use in your classroom or home. If you’d like to share them more widely, please link people here. Thanks.

Fun with tiles

It is no secret that one of my proudest achievements is creating a lovely space on Twitter where people share stories of children’s math talk. Come read along on the #tmwyk hashtag.

That’s where I came across this tutorial-in-photos.

I decided to make myself some. I modified the design a bit (but the food coloring is a genius idea! I used that for sure.)

Then I left them out on Sunday morning and waited for a child to happen along.

Sure enough, Tabitha began making things.

I ate breakfast in the other room.

Ten minutes later, she came in carrying two tiles, put together so that the blue triangles made a square.

Tabitha (7 years old): A square is just a diamond, but I don’t think all diamonds are squares.

Me: Can you draw me a diamond that isn’t a square?

T: The skinny ones wouldn’t be squares.

Me: Yeah. I think I get it. Draw me one, though.

She proceeded to do so. It took a couple of tries.

I lost the paper, but the result looked something like this.

Then, a few moments later she asked a new question.

T: Aren’t all 4-sided things squares?

Me: The doorway isn’t. One of those tiles has four sides but isn’t a square.

I  quickly draw a parallelogram in my notebook.

Me: This isn’t

I drew another 4-sided shape.

Me: This isn’t either.

T: That has 3 corners, not 4. So it can’t be a square.

Me: Show me the three corners.

She counted the three corners that point out from the center of the shape, missing the one that points back inward. She paused.

T: Oh…four.

So What Do We Learn?

Opportunity to think about math is important. Something as simple as leaving an interesting math object out for children to play with can lead to fun math talk.

Tabitha was working on the definitions of square and diamond in this conversation, and she was paying attention to the properties of shapes. This is important work for elementary children. When children are very young—before about first grade—they are learning to identify shapes based on appearances. As they move further into elementary school, they need to start paying attention to properties—the number of sides, the number of vertices (“corners”), etc.

Starting the conversation

Make some of these tiles. The materials cost me less than \$20 (mostly for the wood—I probably could have gotten it a lot cheaper), and the dying and painting took about an hour on a Saturday evening. Then leave them out.

Or leave out a bunch of squares, triangles and rectangles you cut out of construction paper (you can do this for under \$3 and less than 10 minutes of cutting).

Then let the children play and be ready to talk.

Spirals

A few weeks back, this short cryptic video came to my attention thanks to the magic of Twitter.

Thanks to kids connect (@KinderFynes on Twitter)

For more than a year now, I have been posting links and other short bits on Twitter using the #tmwyk hashtag. In the last few months, it has gained momentum. A day rarely goes by without someone posting something interesting or delightful or surprising there.

But back to the video.

We get a very brief glimpse of a classroom of Kindergarteners on a walk. At the moment the video captures, they are trying to decide whether the object on the wall is, or is not, a spiral.

That image in the video was not a spiral because “Spirals are connected”.

So I drew this.

Griffin’s reply: That’s three things connected, not one thing.

So I drew this (sort of).

The part I actually drew was two disconnected spirals. He drew the short line segments on the ends.

Griffin: If you close them off like this, it’s an outline of a spiral.

Next I drew this.

I was wondering whether spirals needed to be roughly circular.

Griffin: In this one, you are looking at a spiral from its edge.

Finally, this one.

I cannot recall his response. We were on the porch on a warm lazy sunny spring morning at the end of a long long winter. We may have gotten distracted.

So what do we learn

This is how I teach critical thinking. Not just at home, but in my work, too. Get the child to make a claim and to give a reason supporting it. Cook up a problematic example and ask for a new claim. Repeat. Quit before angering child.

WARNING: It is my experience with my own children—as well as with my students of all ages—that they learn these lessons well. This means that over time they begin to argue back intelligently, and that they begin to pick apart my own claims and arguments.

Does the Earth have an end?

Talking Math with Other People’s Kids Month continues…

A while back, Rafranz Davis reported a conversation on her blog. She writes frequently about the adventures she has with her nephew Braeden. I asked, and she gave me permission to remix a conversation she and Braeden had about the ends of shapes—especially the ends of the Earth.

Rafranz and Braeden (8 years old) are spending some quality weekend time together when he asks a question.

Braeden: Does the Earth have an end?

Rafranz: Braeden what do you mean by “does the earth have an end”?

B: I’ve been meaning to ask you this question for a long time, at least 2 months. I’ve always wanted to know if the earth stops when you get around it.

Rafranz is a master at the art of mathematical conversation. She asks Braeden a question that gets him talking and thinking.

R: What shape do you think that the earth is?

B: I think that it’s a circle.

R: Really, why a circle?

B: A circle is round.

R: Hmm, interesting. So what shape is that basketball? (The nearby ball may have sparked Braeden’s thoughts)

B: It’s a circle.

B: It’s a triangle.

This is great! Miscommunication. Rafranz is asking about the whole pizza. Braeden is thinking about a slice of a pizza.

R: I mean a whole pizza. What shape is a whole pizza?

B: It’s a circle

R: Why do you think that a pizza is a circle?

B: It’s round and has a center.

R: Earlier you told me that a basketball is a circle and a pizza is a circle. Are they the same?

Again—great move here. Braeden has identified the basketball and the pizza as being round, and therefore circular. Rafranz asks him to compare these two things and to look for differences. She is using Braeden’s curiosity to pursue some deep and important mathematical questions.

B: No, the pizza is flat. The basketball is round…like Earth. The pizza does start and stop when you get all the way around but the basketball can keep going around and around and around.

R: What do you mean around and around and around?

B: If you had a really long string, you can go around the pizza one time but a basketball, you can keep wrapping the string forever. I know why. The basketball is a sphere. (I had no idea that he knew this word)

B: I think that earth is a sphere too and I don’t think that you can go to every single place on earth. I bet that you can keep going around and around and around.

So what do we learn?

Rafranz asks three simple questions at exactly the right moments in this conversation.

1. What do you think?
2. Why?
3. Are they the same?

It turns out that Rafranz really didn’t know enough about Braeden’s original question to answer it the first time around. Those were sincere questions she asked, and they produced a genuine conversation.

Ultimately, Braeden knew that if you walk around the outside of a circle, your path comes to an end—you end up back where you started, having visited all locations on the circle. But if you do this on a sphere, it seemed to him that your path does not necessarily end up back where you started. It’s a lovely insight about the relationship between two-dimensional objects and three-dimensional ones!

Starting the conversation

If you are new to talking math with your kids, don’t worry about getting the timing right. Just start to make a habit of asking those questions. The first few times, you may not get much. That’s OK. It can be like introducing new foods—children need multiple exposures to new things before they accept them. The other question to add to this collection is How do you know?

[Product review] Leap Pad Paint Bucket

Tabitha’s neighbor friend has a Leap Pad. Naturally it became a much hoped-for Christmas gift. She did receive one and has spent quite a bit of time with it.

I have no interest in reviewing the thing itself (although I will give you a heads-up that apps on this thing are expensive in comparison to iOS and Android! Holy buckets!)

The Leap Pad comes with a few standard apps. One of these is a drawing app, called Art Studio. Tabitha (6 years old) has drawn many pictures on it.

This is the sort of thing I’m talking about.

This is fine.

And I wondered whether I could get some math out of it.

See, there is a paint bucket tool in there. When you apply the paint bucket, the paint fills up your drawing, but it doesn’t go across lines you have already drawn. So if you draw a square, you can paint the inside of the square and the paint won’t leak out. Or paint the whole screen outside the square and the paint won’t leak in. Unless you leave a small hole, in which case, the whole screen gets painted because the paint leaks through the hole.

I showed this feature to Tabitha and proceeded to draw some complicated curves, asking her to guess where the paint would go. For example, I drew a spiral.

This was no problem for her.

I asked her how many colors we could use to paint some complicated curve pictures if we used a different color for each section of the drawing.

Again, no problem.

I had her draw pictures and make me guess.

Finally, after about five minutes of this, she announced, “Daddy! You’re not allowed to do math on this!”

I was busted. I had to take a time out and let her just play with her toy.

But then, going back and looking at her more recent art, I can see I got into her head.

Don’t worry, though. The horses are still making appearances.

So if you have a Leap Pad in the house, I gladly give two-hooves-up for math in the Art Studio!