Then I got to thinking about how young children may not have had many opportunities to think about the size of each angle in a rectangle.

Informally, kids will have heard a lot about how a rectangle has “four angles the same size” but if you know that, you don’t necessarily know that any particular size is necessary. Could it be four small angles, or four large angles?

I wanted to design something that would allow children to explore their ideas about that. So I started sketching out some ideas, and the result is a very early draft of a new book.

Imagine you are looking through a circular hole, and you see the blue region that is printed on the next page. What could that blue shape be? Could it be a rectangle? Does it have to be?

In order to answer those questions, it’s going to be useful to think about the angles and the side lengths of shapes you see in your head.

Ideally, a published version of this book will have a whiteboard page with a hole and come with a dry-erase marker so you can sketch your ideas before you turn the page and see a collection of shapes it could be.

For now though, you’ll just want to print the book out, sit down with a child and a pencil and have some fun talking shapes. As always, feedback welcome. Find me on Twitter, or send a note through the Contact page on this blog.

It is undeniable that the pace of posting on this blog has slowed tremendously as my children have aged out of the 4-to-10 year old sweet spot of this work. I have ideas for reformatting and repurposing existing content, and for moving the blog writing forward, and I am hopeful that time will surface for this in the near future.

In the meantime, I wanted to let folks know that I started a newsletter. The first couple of issues went out before we all got sequestered in our homes, and you can read those in the archive. The next one is chambered and ready to go out soon.

So if you’re interested in semi-regular ideas and opportunities related to kids and math, mostly outside of school, then head on over to tinyletter and sign up.

I’ll also put in a plugs here in case you’re looking for something to amuse yourself and/or family with right now. Truchet Blocks are new and available at the Talking Math with Your Kids Store. They are wooden blocks, and so are fun for all ages. They also have some sophisticated geometry built in that allows for beautiful patterns and images.

Note: A link and instructions for downloading and printing a draft of this book are at the end of this post.

It all started with a disagreement between Kassia Wedekind and her daughter Lulu.

Pretty soon, it wasn’t just elevators we were arguing about, and it wasn’t just Kassia and Lulu weighing in. #vehiclechat was born.

With Math On-A-Stick on the horizon, I put together a prototype concept book, and now I am here to report to you what I have learned from many dozens of conversations about the nature of vehicles.

The book begins with some easy, quick decisions. Dump truck? Yes. Salad? No. (Usually. More on this below.) Airplane? Of course! Next up are increasingly controversial cases, including elevator, horse, and broken-down bus with no wheels.

There are about a dozen subconcepts pertaining to vehicles that people carry around in their heads, but which most people have never spoken out loud. Some of these subconcepts are in conflict with each other; some are independent of each other. Some people cling tightly to one or two of them; others loosely hold four or five.

Here are some examples, in no particular order:

Vehicles carry people (not just things).

Vehicles must have wheels.

No living things are vehicles.

Vehicles take people or things from Point A to Point B. (And then there is 4a. Point A and Point B must be different points from each other.)

Vehicles must have an engine, motor, or some other dedicated power source.

A vehicle must have—as its designed and primary purpose—transportation across the surface of the Earth.

Once a vehicle; always a vehicle.

If it could be a vehicle, with a little work or repair, then it is now a vehicle.

One thing we know about math learning is that we nearly always work from examples and mental images of things rather than from logical definitions of things. This is why you can have a second grader tell you in all earnestness that a triangle is a shape with three sides, and also tell you that the image below is not a triangle because it doesn’t look like one.

Knowing a definition (three-sided shape) doesn’t guarantee that you’ll use that definition for naming things. A classic paper by Vinner and Tall details this disconnect as being about concept image and concept definition, and it’s related to Kahneman’s Thinking Fast and Slow. Concept images are the tools of the fast judging brain, while concept definitions are the tools of slow analytical mind.

Somewhere into those controversial cases, there is one that makes everybody think. It’s not the same example for everyone, but each person has at least one. My vehicular conversation partner and I will cruise past airplane and tricycle, and then they will pause and smile and look off into the middle distance. That moment is the one I seek—when you notice that your quick reactions aren’t good enough and now it’s time to think—even if only for a brief moment.

Also I have learned that metaphorical examples are more comfortable for many people than close, but not-quite-right examples. This means that I’ll often hear a quick “Yes” to salad (a vehicle for nutrients into my body), but a slow “Yes” or even a “No” to elevator (because it doesn’t have obvious wheels, or has a limited range of travel, or for some other reason).

Most of all, I have learned that these conversations are really a lot of fun. People smile. They laugh. They think of additional challenging examples. They ask “What about?” and “What if?” They imagine what other people might say, and why particular examples might be controversial.

This is a social game of negotiating meaning, and of noticing that language which seems so precise is often not that at all. Play along with us! Offer up some examples in the comments, or some additional vehicle subconcepts not listed above. Or join the conversation on Twitter.

Here’s a link to download a copy of the book. You’ll want to print two-sided, FLIPPING ON THE SHORT SIDE, and then trim top and bottom with scissors or a paper cutter. The book then assembles tidily as a 5.5 inch square.

And if you like books in development, check out a fun new book about shapes: What Could It Be?

A new version of How Many? is coming out on Tuesday, from Charlesbridge. It has an all new design, some new words, and mostly the same images, but is published for the home market. (Whereas Stenhouse publishes primarily for the teacher professional development market, and so is still the place to pick up the teacher guide.)

Charlesbridge did a lovely job with Which One Doesn’t Belong? earlier this year, and this one has come out beautifully as well. Both have been well reviewed in Kirkus Reviews.

And we’re celebrating with a launch party at the Red Balloon here in Saint Paul, MN. Tuesday, September 10 at 6:30 p.m. (The Red Balloon is on Grand Avenue, just west of Victoria.)

We’d love to have you join us! We’ll count and play with math toys and maybe even chat about vehicles!

Of course they blew me away. Perhaps a favorite moment was when a fifth grader told me that each turtle made space for another turtle, and that also each turtle needed the other turtles to make space for it.

Across the images, these students saw repeating and growing patterns. They saw patterns involving shapes, colors, numbers, and combinations of all of these.

I encourage you, the reader, to take some time with some children and ask each other these questions:

What patterns do you see?

So what repeats?

How does it repeat?

Then report back. Let’s learn together.

In the meantime, you should know that a new version of Which One Doesn’t Belong? came out this week. Same shapes as the original, new book design, new colors, and a few new words. Importantly, this is the first in a series of collaborations between Charlesbridge and Stenhouse by which a home version of each Stenhouse book is issued by Charlesbridge about a year after first publication.

Look for it (or even better…ASK for it) at your favorite bookseller.

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.)

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.

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.

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.

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,

21st Century Pattern Blocks

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.

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!

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/