Public Math

This is Molly.

This is Chris.

Molly is a math leader who lives in Vancouver, WA and started a project called Math Anywhere. Chris lives in Chicago where he supervises secondary math for the Chicago Public Schools and leaves math in places for others to find.

I met each of them through the work of this blog, and through Math On-A-Stick. We agreed that it would be fun to have some together this summer to brainstorm and prototype new forms of creative, playful math engagement for people in unexpected places.

One thing to know about me if we ever have the chance to work together is that I don’t work small or slow. One year I got the idea for Math On-A-Stick; the next year it was up and running at full scale at the nation’s second-largest state fair.

That tendency of mine turned an afternoon with Molly and Chris into a 48-hour extravaganza we’re calling the Public Math Gathering. With the generous support of Desmos and CPM, we added seven more amazing folks to our guest list and got started planning.

This is Morgan.

Morgan teaches at South High School in Minneapolis and runs a math circle at the Rondo branch of the St Paul Public Library each summer. We brought Morgan on board to help us plan and execute this work.

This is the whole team, including the four named above and our friends Aeriale Johnson, Manju Connolly, Lamia Abukhadra, Andrés Lemus-Spont, Amy Nolte, and Lara Jasien.

Together, this team includes designers, artists, elementary and secondary teachers, researchers, writers, and teacher educators.

Here’s how we have spent our time.

On Friday evening, we walked the boundaries of the Midway, Macalester-Groveland, and Frogtown neighborhoods of Saint Paul looking for opportunities for mathematical provocations. Here are some things we noticed and talked about.

 

On Saturday, we volunteered as a group at Math On-A-Stick, and then went off-site to do two pieces of work:

(1) Break down and critique our experiences at Math On-A-Stick, and

(2) Design new math provocations for new contexts.

Our reflections on Math On-A-Stick involved these questions:

  • Who was there?
  • Who was not there?
  • How did you notice those who there engaging with the materials and the space?
  • What are the affordances of the Minnesota State Fair as a place for designing math interactions?
  • What are the limitations?

Our design work followed from this invitation:

Let’s design place-based opportunities for [X] to engage in math provocations that are [Y]. 

We filled in many possibilities for X and for Y. For some of us, X might be children or teenagers or families or the math anxious. And Y might be delightful or memorable or identity-changing or surprising.

We moved on to prototyping, and we’ll test drive some new ideas at the Fair on Sunday morning.

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?

Counting in downtown Saint Paul

I had my first book event today—for How Many? at Subtext Books in downtown Saint Paul. Lovely people, great little independent bookstore. You should buy some books from them.

We had a small but loyal crowd that included a three-year old and an eight-year old. The three-year old was charming, as all three-year olds are, and today she answered all yes/no questions in the affirmative. She and I talked about shapes and eggs and money. It was good times.

But I really got to get into the head of the eight-year old.

We discussed the grapefruit page below, and the unsolved mystery of whether there are exactly six grapefruit—the ones we can see directly—or more than that with at least one hiding underneath, possibly reflected in the surface of the bowl.

6.jpg

We moved on to the next page, which is where the real fun began.

7

My eight-year old conversation partner looked carefully, thought for a while, and announced that there must have been more than six grapefruit on the previous page because there are more than six on this page.

I asked, “How do you know?” and it turns out he was visually pairing the grapefruit halves on this page. He used his fingers to show me the pairs he made, but he was having trouble keeping track of their number. So when he came out with more than six pairs of grapefruit halves on this page, that meant there must have been more grapefruit in the bowl.

We flipped pages back and forth several times while sorting this out, and he finally concluded that there were six grapefruit on both pages. Children rarely have math tasks that connect this way, but they expect that the tasks should connect. It was delightful to watch this expectation play out.

Next up was the avocado page.

10

He thought for a bit and decided there were “seven point five avocados”. I thought I knew how he knew—same as the grapefruit—but I asked to be sure, and I was wrong.

“Three fives is fifteen, and then divide by two.”

It took a few more exchanges to extract that dividing by two makes sense here because there should be half as many whole avocados as there are half-avocados. Of course this is brilliant and important mathematics, and it arose in the context of making sense of a meaningful counting situation. Also notable is that three fives was a fact he retrieved quickly while three fours (of grapefruit halves) did not seem to occur to him.

The lesson here is that children are brilliant. They build math out of their everyday experiences, and when you offer them opportunities they apply the math they know to make further sense of their worlds.

Another lesson is that my new book—titled How Many?—is out. The best price and free shipping are at Stenhouse.com. If you read it with children, please report back and maybe leave a review at Amazon.

 

Cocoa Puff or Cocoa Puffs: The language of nothing

In honor of Tabitha turning 11 this week, here’s a conversation from 6 years ago. 

We have a little family tradition. When we go grocery shopping the weekend before your birthday, you can choose one box of any cereal you want-no restrictions. In the weeks and months leading up to the grand event, much time is spent in the cereal aisle weighing the advantages of the various sugar-laden options.

The week before turning five, Tabitha nearly dropped the ball. She just grabbed the first box of anything at hand. I don’t remember what it was, but it seemed out of character for her. I reminded her of the cereals she had been coveting as recently as the previous week.

She went for the generic Cocoa Puffs.

I steered her towards the real deal. If you’re only gonna eat ’em once a year, you might as well have the sugar-addled bird bouncing off the box in front of you, right?

A sugar-addled bird

One morning shortly afterwards, we had this conversation:

Tabitha: Do I have Cocoa Puffs or Cocoa Puff in my hand?

Me: Well, you have four Cocoa Puffs.

T: [with only one in her hand now] Do I have Cocoa Puffs or Cocoa Puff?

Me: You have Cocoa Puff.

T: [huge smile] Right!

Me: [with empty hand displayed] Do I have Cocoa Puffs or Cocoa Puff in my hand?

T: [silent but smirking]

Me: Well…Is it Cocoa Puff or Cocoa Puffs?

T: [continued silence]

Me: I have zero…

T: [bigger smile]

Me: …Cocoa…

T: Puffs!

Me: Yeah. Isn’t that weird? If you have one, it’s Puff; if you have none it’s Puffs.

T: I knew that.

Me: Of course you did.

T: No! I knew that; I was showing you that [you had zero] by not saying anything-zero words!

So what do we learn?

Children listen carefully to language patterns. They do not learn a native language like a second language in school. The rules are not carefully explained to them one at a time.

Instead they listen, speak, get corrected, and try again. All of this can be tremendously fun for child and parent alike.

It is an odd quirk of English that zero is plural, grammatically speaking. We talk about having one child, but zero children. More commonly, we use no instead of zero, as in My neighbors have no children. The grammar is the same either way; saying My neighbors have no child sounds funny to our ears.

Starting the conversation

In discussing place value, zero is sometimes called a place holder. To understand that, children need to understand zero as a number. They need to understand that zero can legitimately answer the question, How many are there?

We talk a lot about zero in our house. You can too. Ask your children, “Would you rather have one cookie, two cookies or zero cookies?” Ask who has more of something, even when one of the people has none.

The Mad Hatter in Alice in Wonderland gives an example of this. The March Hare offers Alice “some more tea”. When Alice says she can’t possibly have more, since she hasn’t had any yet, the March Hare replies, “It’s very easy to take more than nothing.”

Another silly language game we play in our house is this. If you look in the pantry and see that there are three cookies left, you can report this in the following two ways: (1) “I checked the cookies; there are three left,” and (2) “There are three cookies.” If, however, there are no cookies in the pantry, these two ways of reporting the sad fact become: (1) “I checked the cookies; there are none left,” and (2) “There are none cookies.” We like to treat none as a number. There is no good reason for this; it is for personal amusement purposes only.

Postscript

Tabitha again chose Cocoa Puffs on this, the week of her eleventh birthday. She is enjoying them, but she has also stated the obvious—they look like rabbit poop.

Things that give me hope

I am excited to see more and more people working hard to connect students’ informal mathematical thinking to the more formal work of schooling.

The emphasis in the school-home relationship used to be on helping kids do homework (as parodied in these five seconds of the newest Incredibles trailer).

No more! These days there are plenty of projects that seek to stimulate children’s math minds in ways that parallel what we do with literacy.

I’m thinking of the beautiful work of The Museum of Mathematics in New York City, and of Dan Finkel’s Prime Climb and Tiny Polka Dots games. I’m thinking of Malke Rosenfeld’s work, and of Bedtime Math and their associated research at the University of Chicago. and I’m thinking of Table Talk Math.

I’m thinking also of Eugenia Cheng, whose How to Bake Pi does for adults what I want parents to do for kids—show how their natural ways of thinking about their everyday worlds are deeply mathematical.

Some of the momentum for these projects can be traced back to the Cognitively Guided Instruction (CGI) research at University of Wisconsin, which demonstrated that when teachers know the informal ideas about numbers and operations that kids bring to school, those teachers are more effective at helping students learn the formal mathematics of school. The copyright on the first CGI book—titled Children’s Mathematics—is 1999, and it documents research that had been going on for some time before that.

Many of us doing this work now are deeply influenced by this work. Progress on this sort of thing is slow. Time spans are measured in decades, not months or years. But it’s a vibrant space that’s growing. I am optimistic.

Now for the point of today’s post. I want to recommend a delightful new book, Funville Adventures by A.O. Fradkin and A.B. Bishop, and published by Natural Math.

Funville Adventures involves a series of characters in a fanstatical land. Each has a magical power; these powers interact. You think you’re just following some fun and silly adventures on the playground; really, you’re thinking about one of the most important ideas of higher mathematics—functions.

Yet true to the nature of most of the projects I discussed above (and to the nature of this blog), it doesn’t matter whether you know about the relationship between the story and the mathematics. If you do, that’s great. If you don’t but are curious, there’s an addendum for that, and if you just want to stay at the level of the story, you’ll exercise your math mind thinking about the relationship between growing and shrinking, the relationship between doubling and halving, and why flipping upside down has no sibling.

Funville Adventures should be in every Talking Math with Your Kids-friendly library. I supported it as a Kickstarter; I’m a big fan of A.O. Fradkin’s blog. The book is on sale right now. More info and reviews here.

Time Zones

Griffin is 13 years old and seems to be coming to the end of that early adolescent phase of rejecting everything those around him hold dear. Engaging him in math talk has taken more finesse in this phase of life.

Mostly it has involved giving him responsibility for things that involve making calculations. When he was little, we could talk collaboratively about how many tangerines are in a 3 pound bag and discuss whether this would be enough to last the family a week. Now I tend to put him in charge of getting enough tangerines to last us a week. He still has to do the same thinking, but he’s in charge.

This is not enough tangerines for a week at our house. (By the way, which is more?)

From time to time, though, we still put a mathematical idea up for discussion, and as he ages through adolescence, these conversations happen a bit more often. Yet he is still wary. Nevertheless, I persist.

We have been watching the Olympics, and we have wondered about which events are happening as we watch them, and which ones happened earlier (yet somehow happened “tomorrow”!)

Griffin was thinking about time zones, and about their implications for traveling as we wrapped up an evening this week, and made preparations for the next day.

Griffin (13 years old): So they’re 14 hours ahead of us?

Me: Yes.

G: You’d get a lot of jet lag, huh?

Me: Yeah. Maybe not as much as it looks like, though. Maybe it’s just 10 hours’ worth, going the other way.

There is a bit of a puzzled silence.

G: Wait. Really?

Me: Yeah. Well, plus a day.

G: Wait. Is this one of your mathy talks?

Me: Absolutely not.

If you’re reading this, Griff, I’m sorry (sort of). I am totally busted.

Me: Yeah. 14 hours ahead is the same as 10 hours behind, right? Just going the other way.

G: But the day would be wrong.

Me: Yeah. You have to add a day, but you don’t get jet lag because the day changes, you get jet lag because the time of day does.

G: Maybe.

He returns to packing his lunch. I go back to whatever I was doing. Putting turtles in boxes, probably.

A couple minutes later…

G: So the east coast is 23 hours behind us?

So What Do We Learn?

Keep trying. Opportunities to talk about numbers, shapes, and patterns present themselves. Seize them and do not stop. Ask questions, think out loud. Don’t worry about whether any particular conversation goes anywhere. Just keep at it.

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.