In Defense of the Diamond

I had an opportunity to play math with children and parents at a local elementary school last night. (Shoutout to Oak Ridge Elementary! Thanks for hosting!) The Pattern Machines were a big hit.

One seven-year old spent some time deciding what to make and a couple of minutes idly punching buttons before she got down to work. In a short time, she declared she had made a square and presented it for my approval.

I noticed her square, as this seemed important to her, and I told her it gave me an idea. On my own Pattern Machine (leaving hers untouched—an important tenet of this work is never to take the pencil out of the child’s hand!), I made the thing below and asked her whether I too had made a square.

“No. That’s a diamond,” she declared.

I rotated my Pattern Machine 45°.

“Now?” I asked.

Now it’s a square.” she said with a knowing look.

Diamond also came up when I talked with children and adults about a page in Which One Doesn’t Belong?

I used to think that diamond was a lazy term for rhombus, but it is not. Diamond has a stable and robust meaning that is different from rhombus. On that Which One Doesn’t Belong? page, the upper right and the lower left shapes are diamonds. The lower right is not, yet each of these three is a rhombus.

Also sometimes a pentagon is a diamond.

So I propose we treat diamonds as we do other mathematical objects.

Let’s build rock solid definitions of them—definitions that we can take as shared and use to sort diamonds from not-diamonds.

Let’s investigate the consequences of those definitions.

Let’s investigate conjectures and prove theorems.

Together, let’s build a rich field of mathematical inquiry.

I’ll start us off. Some diamonds can be cut into smaller diamonds, as in the example below. Can ALL diamonds be cut into smaller diamonds? If not, which ones can and which ones cannot?

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.

6

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? 1 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.

Some additional prompts

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.

Math blocks how-to 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.

Tabitha making a zig-zag pattern with the math blocks

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.

Skinny diamond

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.

Non-rectangular parallelogram

Me: This isn’t

I drew another 4-sided shape.

Concave quadrilateral

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.

I decided to ask Griffin (9 years old) about this to see what his ideas would be.

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

So I drew this.

spiral.post.1

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

So I drew this (sort of).

spiral.post.2

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.

spiral.post.3

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.

spiral.post.4

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.

R: What about a pizza?

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.

6.slice.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)

R: What about Earth?

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.

lf_art_000003This 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.

lf_art_000085

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.

lf_art_000086

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.

lf_art_000070

 

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

 

 

[Product Review] Tupperware Shape-O Toy

You know this thing.

This thing has been around for many, many years. You may not know that it is officially the “Tupperware Shape-O Toy” but if grew up in or near the United States anytime since about 1960, you have encountered this toy. It is the rare math-y toy that is actually awesome in the ways it was intended to be.

(See discussion of the Multiplication Machine on this blog for an example of a math-y toy that is awesome in unintended ways. See your local Target for a wide selection of math-y toys that are not awesome in any way at all.)

We had some fun on Twitter last fall when a math teacher and father, Dan Anderson, invited speculation about which shapes would be easiest and most difficult for his 1 \frac{1}{2} year-old to put in the holes.

[Fun fact about Dan Anderson—if you heard last year about how Double Stuf Oreos are not actually doubly stuffed, it was his classroom that got the media ball rolling.] Anyway, here is his ranking—following his son Calvin’s lead:

And here is an amusing video of a cute kid playing with one. The parental participation in the play may be a bit heavy-handed but the spirit is right—encouraging and playful.

Notice that the triangle is harder for him to fit in than the square, and that it’s tough for him to distinguish the hexagon from the pentagon.

Tons of fun to be had with this classic!

 

Short notes

I passed a lovely morning with some math teacher friends at the Museum of Math in New York City today. Sadly, I did not have Griffin or Tabitha along. We would have had a ball.

I have three observations…

One. There is lots for kids to do there. It is truly a bonanza of mathematical conversation starters. I left wanting to be hired on as “Docent for Talking Math with Your Kids”. This is because the math does not smack you in the face. Instead, the math in the exhibits tends to surface in the process of playing, experimenting and paying very careful attention to what is going on.

In short, you want to talk about these exhibits and you want to linger.

Two. Many of the exhibits are designed in ways that allow kids to play and to do math beyond the intended activity. The exhibit below, for example, kept two little boys (each 5 or 6 years old) busy for 20 minutes although the original activity was way beyond them.

The first five minutes, they were sharing each of the shapes equally, making sure each of the two boys had the same number of squares, and the same number of triangles, et cetera. The next 10 minutes, they spent arranging the shapes on the screen, marveling at the things this provoked on the screen beneath.

It was lovely.

It could have been a bummer, though. If someone had insisted on doing the intended activity, these kids could not have done it. If the shapes had been electronic instead of concrete, the kids would not have been able to play with them in such creative ways.

Well done, Museum of Math!

Three. On the subway train back to my hotel, I noticed a little girl—probably 4 years old—holding up four fingers as she sat between her mother and brother.

I moved to be in listening range.

It turns out, she had been told they would be on the train for five stops. She had been putting up one additional finger as the train left each stop.

Sister (4 years old): [She is holding up four fingers as the train enters the station] This is our stop!

Brother (8 years old): No. The sign says there’s one more stop.

Sister: But you said five stops.

Mom: One more, sweetie.

The train stops. People get off and on. The doors close. The train starts up again.

The girl keeps exactly four fingers up the whole time.