Which Poster Doesn’t Belong?

(Cross-posted from Overthinking My Teaching)

Two and a half years ago, I was developing Which One Doesn’t Belong? (before Stenhouse had signed on to publish it). I went on a tour of elementary classrooms to talk with K—5 students all around the Twin Cities about these collections of shapes. I learned a tremendous amount of course, and much of that learning went into the Teacher Guide (which Stenhouse convinced me needed to exist).

I learned a lot, and I also noticed something.

Most of those classrooms had some form of shapes posters on the walls. Triangles, rectangles, squares, and rhombi were proudly and prominently displayed so that students would be surrounded by correct geometry vocabulary. Most of those shapes posters had something important (and unfortunate) in common with the shapes books in the school library and in the children’s homes.

There were rarely squares on the rectangle poster. All of the triangles were oriented with one side parallel to the ground, and most of them were equilateral. Sometimes the shapes had smiley faces. You and I know that a triangle is still a triangle, no matter its orientation. I can assure you not all elementary school children know this. While the vocabulary is good on your standard shapes poster, the math is not. (I decided not to link to examples—you can do your own search and report back if you find my claims exaggerated.)

This summer, Stenhouse is helping all of us to fix this. You can now preorder Which One Doesn’t Belong? shapes posters.

Poster images

They come as a set of eight, with an insert in the spirit of the Which One Doesn’t Belong? Teacher Guide to help you facilitate student thinking and classroom conversation as they hang in your classroom.

1. Which SQUARE doesn’t belong?

2. Which RECTANGLE doesn’t belong?

3. Which RHOMBUS doesn’t belong?

4. Which HEXAGON doesn’t belong?

5. Which TRIANGLE doesn’t belong?

6. Which POLYGON doesn’t belong?

7. Which SHAPE doesn’t belong?

8. Which CURVE doesn’t belong?

These posters are filled with good mathematics. Consider the triangle poster on top of the pile. The triangle in the lower right is the only right triangle. The one in the upper right is the only equilateral triangle. The one in the upper left is the only isosceles triangle (or is it? do equilateral triangles count as isosceles?) The one in the lower left is the only one you can’t build out of the triangle in the lower right. Students will notice side lengths, angle measures, orientation, composition and decomposition, and more properties of triangles. Some will complain that not all of them are triangles (“too pointy” or “doesn’t have a bottom”). These posters let you and your students sit with—and play with—these ideas over a period of weeks or months.

So as you plan your back-to-school classroom organizing and decoration, I hope you’ll consider making space on your walls for these posters. And I definitely hope you’ll share your students’ ideas here and on Twitter.

Available for pre-order now. They’ll ship in early August.

Young children are more logical than you think they are

The kids (Griffin—12 years old—and Tabitha—10 years old) and I went on our annual summer camping trip recently.

This year it was Glacial Lakes State Park in western Minnesota.

While there, we visited the swimming beach where a family that included a 3-year-old girl was also enjoying the beautiful warm day.

There were minnows swimming in the shallow water, which the 3-year-old badly wanted to capture with her net. When she stood still, the fish would approach cautiously, but every time she moved, the fish quickly bolted.

Girl (three years old): I want to catch them, Mommy! Why do they swim away?

Mom: Maybe they’re scared.

[Thoughtful pause]

Girl: But I am not a monster! I am not a monster, Mommy.

Over the next several minutes, she repeated her monster claims a number of times. Eventually—as will happen with 3-year-olds—her attention shifted to other things and the swimming and splashing continued.

So What Do We Learn?

Young children can think logically. This runs counter to some assumptions we make about three-year-olds, but it is true.

Here is the logical truth this girl understood:

The only thing to fear is monsters.

Fish fear me.

Therefore, I am a monster.

And deeper yet, she understood what logicians call the contrapositive.

The only thing to fear is monsters.

I am not a monster.

Therefore, the fish do not fear me (and so I can catch them).

This child was not expressing horror at being considered a monster. Rather she was a little frustrated that the fish were not behaving according to the logic she knew to be true. Or perhaps she was a little frustrated with the inadequate (and illogical) explanation her mother had provided.

In any case, her logic was perfect.

We don’t really expect this of three-year-olds but we should. Just as we don’t really expect rich place value ideas in kindergartners, but we should. If we keep our ears and eyes open, we’ll see it and hear it and be able to support its growth.

A delightful new book on Kickstarter

There are a bunch of people doing really good and interesting work with math and kids these days. Sasha Fradkin is one of these. She has a gift for tapping deep into kids’ mathematical minds and for writing about the beautiful ideas she finds there.

She has written a book—Funville Adventures—that is definitely worth your time and money, and she’s funding its publication on Kickstarter. I have pledged. You should too. I promise you’ll be glad you did.

National Math Festival

I’m taking Math On-A-Stick and Which One Doesn’t Belong? on the road—to the National Math Festival in Washington, DC on Saturday, April 22, 2017. If you’re nearby, you should come out and say Hi!

I have two Math On-A-Stick sessions—at 10:00 and 4:30—and one Which One Doesn’t Belong? session at 2:30. The Math On-A-Stick sessions are pure play; the Which One Doesn’t Belong? one is interactive but more talky.

There are lots of other amazing folks doing amazing things with all ages, so come spend the day! It’s free. See you there.

The Summer of Math is back!

We had so much fun the first time around, the Summer of Math is back for a second year, and it has its own webpage!

The basics are the same as last year:

You can head over to The Summer of Math webpage, pay for a subscription, and all summer long (June—August) we’ll ship you awesome, fun stuff that will keep you and your 5—10 year old(s) busy playing and talking math.

You’ll color, count, make patterns, designs and shapes. You’ll read together, draw, and challenge yourselves. You’ll notice. You’ll wonder. You’ll play. And when school starts back up in the fall,

your kids will remember this as the best, mathiest summer ever.

A few small changes include: a new website, three boxes instead of four (but same amount of total mathy goodness, so the net result is higher math-fun density in each box), and a few new things rotated into the lineup (the Truchet tiles are curvy this year!)

Together with a little help from some super smart friends, we’re shooting for a bigger Summer of Math this year, but there is an upper limit on subscriptions. Help us reach it—and as many families as possible—by signing up and by spreading the word.

Fractivities! Coming soon to a school near you! (If you issue the invitation.)

New to Minnesota, Jennifer Schuetz from the Fractal Foundation brought fun fractal activities, or fractivities, to Math On-A-Stick last summer.
Younger students had the opportunity to learn how the Sierpinski triangle is a fractal – by repeating a simple pattern over and over again, smaller versions of the same pattern are created.
Older students created their own Sierpinski triangles with wooden sticks and glitter glue to make a fractal with sticks!
With help from Minneapolis High School art teacher Stephanie Woldom and math teacher Morgan Fierst, we had tons of fractal fun with hundreds of children and their parents from across Minnesota!
Jennifer’s dream turned into reality: job title of visiting artist/mathematician!

Jennifer leads fractal education in classrooms and other venues across Minnesota, the U.S. and the world (her geographic reach is ever-expanding just like fractals!). Fractals are not only appealing to children but also adults… even senior citizens have fun learning about them!

Instructions to 12 fractivities and associated worksheets and answer keys are at: http://www.fractalfoundation.org/resources/fractivities.

What do you think about projecting videos of fractal zooms accompanied to original music onto the dome of a planetarium? The Fractal Foundation also does this! More information can be found at:  http://fractalfoundation.org/fractal-shows/fulldome-content/

You can also get in touch with Jennifer to see how to do this. Check out some videos at: http://fractalfoundation.org/videos/

Finally, Jennifer is looking for gigs in schools! Get in touch through the Fractal Foundation Facebook page.

Cold snap

Tabitha (9 years old) is keenly attuned to the temperatures these days, as subzero air temperatures or wind chills mean indoor recess. Being a child of great physical energy, indoor recess is not ideal.


We have an indoor/outdoor thermometer on our kitchen table, which she checks several times a day. Yesterday evening before doing the dishes together, she checks the thermometer.

Tabitha (9 years old): It’s 1 below.

Me: What was it this morning? Five degrees?

T: Four

Me: Crazy. So it’s colder now.

T: Yeah.

Me: How much colder?

T: Five below

Me: How do you know? Is it because 4 + 1, or did you count?

T: Neither.

Me: Oh! Now I have to hear it!

T: Well…Four minus four is zero, then it’s one less, so it’s five.

Me: So one more than four less…er…one less than four….no….

[we laugh]

T: It’s one more because it’s one less!

So what do we learn?

This conversation reminded me very much of a game I used to play with Griffin (who is now 12 years old) on cold winter mornings. In both cases, the children naturally developed a strategy using zero as a stopping point in making comparisons.

The thing I especially love about this story is that Tabitha expresses a complicated relationship that is crystal clear to her: “One more because it’s one less.” Expanded out, she’s saying that “The difference between -1 and 4 is bigger than the difference between 0 and 4—the difference is bigger by 1 because -1 is one unit further from 4 than 0 is.”

She can express this complicated idea because it is her own.

If I tried to tell her that this is how subtraction with negative numbers works, she would definitely pronounce my ideas confusing—whether they were expressed in the language of 9-year-olds or the language of mathematicians.

I cannot tell her these things and have them be meaningful. What I can do is ask how much colder it is now than it was this morning.

Starting the conversation

Move to Minnesota.

I’m kidding.

You can buy a Celsius thermometer, though.

You can make comparisons more generally, both asking your child how she knows, and talking about how you think about it. How many more full cups in the muffin tin than empty ones? How many more fork than spoons? How many more adults on the bus than children (or vice versa)? How many more quarters than dimes in the change bowl?