# 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.

# Tessalation: A great new book

Tessalation is a terrific new picture book by Emily Grosvenor. The story involves a little girl whose mother needs a bit of peace and quiet, so sends her outside to play. While outside, Tessa (get it?) notices shapes fitting together without gaps everywhere she looks.

I helped sponsor Tessalation on Kickstarter this spring, and our hard copies arrived last week. Naturally Tabitha (9 years old) and I read it together right away.

Here are some of the things Tabitha, Griffin (11 years old) and I noticed and discussed while reading it, and afterwards:

• The turtles are delightful.
• While they are somewhat different turtles from the ones we’ve played with around the house for the last year, they have an important characteristic in common—two noses and two tails come together in both tessellations.
• There are tessellating leaves that look an awful lot like some shapes I’ve made and we’ve played with a number of times. We saw kites and hexagons and triangles in the leaves just as we have in the pink quadrilaterals below.
• We wondered whether this object counts as a tessellation. (It’s not from the book, but Tessa set a great example for us to notice and ask about tessellations in our world.)

All in all, Tessalation is perfectly aligned with the Talking Math with Your Kids spirit. It creates a richly structured and playful space for parents and children to notice things and to converse. The language is fun. The images are beautiful. Tabitha and I highly recommend it.

Quick notes: Tessalation will be a component of August’s Summer of Math box. It’s not too late to sign up! Also, we’ll soon have a Tessalation/Tiling Turtles combo pack available. You can order the book right now from Waldorf Books, and e-books from Amazon.

# [Product review] The bathtub

Talking Math with Other People’s Kids month continues…

Today we pay tribute to the family bathtub, and its profound contribution to family math talk over the centuries.

Don’t laugh! Is yours more perfect?

Dad and loyal reader Jon Hasenbank reports some math talk at bathtime with his own 5 year old son, whom we will call Isaiah.

Isaiah is in the bathtub, having a lovely time. He has stacked his bath-toy Elmo on top of his bath-toy Cookie Monster.

isaiah (5 years old): Look! His eyes are peeking out!

He did not report further details to me.

But he did demonstrate an important principle of talking math with your kids—It’s not a conversation until you, as a parent, participate. When Jon turned Isaiah’s observation into a wondering, he set the stage for some good math talk.

The bathtub is great for this!

Tabitha has complained about the depth of her bath in the past—always that it is not deep enough. “It’s not even one foot deep” she has wailed as her toes stick out of the water. “Is it one hand deep?” I have asked. And—as with Jon and Isaiah—we have been off and running on a lovely exploration of measurement.

# [Product Review] Candy Mega Buttons

This is our first audience-participation post.

I bought these at the Minnesota State Fair last summer.

When you open the package, here is what is inside.

(Click for larger version of this image, which you are free to download.)

I am curious how my readers would use these to talk with their children. Please feel free to post hypothetical as well as actual conversations in the comments.

There is no one right answer for this activity. See what fun you can have with them in your home, and report back!

# [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!

# [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!

# Multiplication Machine [Product review]

We have in our house a thing called the “Multiplication Machine”. It is a flash-cardy sort of thing. The multiplication facts are written on top of a 9×9 array of spring-loaded buttons. You press one and the button pops up, revealing the product. You can buy such a thing at your nearest teacher supply store. Ours came from Lakeshore Learning.

Talking Math with Your Kids is dedicated to helping parents and other caregivers to identify the mathematical opportunities afforded by everyday life, so we will not discuss here the traditional, intended use of this product (which is drilling and reviewing multiplication facts).

Tabitha was hard at work pressing buttons on the Multiplication Machine the other evening. When I peeked in on her, I saw a scene that looked an awful lot like the one below.

She was playing with the arrangement of up and down buttons, not with the multiplication facts written on them. Patterns are tons of fun. So I went with it.

We developed the up, up, up, down pattern. We went across each row from left to right, top row to bottom row, as you would read a book.

We developed its opposite—Down, down, down, up.

We developed the Up, up, up, up, down pattern. This proved much more difficult for Tabitha, as she could not subitize the four ups. She counted them on her fingers, which she also needed for pressing buttons. She worked it out, though.

Before executing this last one, we noticed the right-to-left diagonals we had gotten from the Up, up, up, down pattern and predicted what we thought would happen when three ups became four. She correctly predicted the left-to-right diagonals, but I do not know why she predicted this.

We have not yet investigated the down-down-up pattern together, but I suspect she will get a kick out of it.

There are many more cool patterns to play with here. A few ideas that I am sure we’ll explore in the coming weeks:

• What will happen with lots of different combinations of ups and downs?
• What if we do columns instead of rows?
• Are there any patterns where you cannot tell whether the person did rows or columns?
• What if we follow a path back and forth across the rows, instead of starting at the left-hand side of each row?
• What if go right-to-left? Or bottom-to-top?

# Talking math with a word game [Product review]

Long ago, we were given the game “What’s Gnu?” as a gift.

What better time to dig it out than in the waning days of summer leading up to the return of school? So Tabitha (six years old) and I did just that a couple weeks back.

Neither one of our children has been an early reader. They both love books. They are highly verbal with substantial vocabularies. And neither one has ever wanted to read aloud.

But “What’s Gnu” is all about reading words aloud. This presented a problem.

So I got creative.

We played “War” with letters.

See that green mechanism to the right of the box in the picture? That is a letter dispenser. You move it back and forth, dispensing two letters at a time.

In our game, we took turns dispensing (it’s totally fun). Tabitha’s letter was on the left each time, mine was on the right. Whoever had the letter that came first in the alphabet won the round, taking both letters. Largest number of letters at the end of the game wins.

We discussed strategies for knowing who won each round. Tabitha described some version of each of the following strategies.

1. Middle/end. Example: M and X. M is in the middle of the alphabet;  X is at the end. Therefore, M comes first.
2. Recite alphabet from the beginning. This comes in two versions: (a) stop reciting at the first of the two letters, and (b) stop reciting at the second of the two letters. This one is useful for two letters that are in the same part of the alphabet. Example: H and N (both can be seen as in the middle of the alphabet).
3. Recite alphabet from the letter you think is first. This is a more efficient version of strategy 2. Example: L and P.
4. Adjacent. When two letters are next to each other in the alphabet, you can know right away. This may just be a very quick version of 3. Example: H and I.
5. ABYZ. These letters are so close to the beginning (or end) of the alphabet that they MUST be first (or last), no matter what the other letter is.

There are relationships to something called subitizingwhich refers to knowing how many things there are without counting. You can recognize three objects, and probably also four, without counting or grouping them. But five objects you cannot; you probably group them as three and two without even noticing it. How psychologists measure this fact is super-interesting but not pertinent here.

Instead, notice that strategies 4 and 5 above are like that. Tabitha could recognize adjacent letters without thinking or reciting the alphabet. Reciting the alphabet is like counting. Similarly A, B, Y and Z she could compare to other letters without reciting.

But here’s the point. This counts as talking math.

We were comparing the order of things. Letters, like numbers, have an order. Anytime we are talking about how we know what order things come in, we are talking math.

I did mention that product reviews would not take the usual form on this blog, didn’t I?

# Waffles [Product review]

From time to time, we will be reviewing products here at Talking Math with Your Kids. Sometimes they will be products that are intended to foster mathematics learning, but not always.

Today, we consider one that is not.

We recently bought Kellogg’s Eggo Homestyle Minis waffles.

Tabitha is obsessed with waffles. We typically get the store brand, which as far as I can tell are mathematically uninteresting. But every so often Eggos go on sale, and then it’s game on!

Consider the Minis.

The minis come in sets of 4.

Here is the kind of fun we can have (and, I assure you, that we have had) with this:

• Say, “I’m making frozen waffles this morning. How many do you want?” Leave the unit deliberately unspoken. Child says “one” and is served one mini waffle. Discuss.
• Do the same thing again the next morning.
• Hold up a set of four waffles and ask a young child (say, 2 to 4 years old) how many you have (answer is likely “four”). Then point out to the child that it says there are “10” in the box. Dump them all out and discuss. Key question: What are there ten of?
• Ask a somewhat older child (say 7 to 9) “If there are 10 sets of 4 waffles in the box, how many waffles are there?” Follow up with “How do you know that’s right?”

Finally, this: In addition to (1) waffles, and (2) sets of waffles, there is a third unit to count in that box: servings.

It turns out that 1 serving is 3 sets of 4 waffles. How awesome is this?

You can ask an older child to predict what the number of “Servings per Container” will be on the Nutrition Facts label. I would have gotten it wrong. I would have applied too much mathematics to the problem and said $3\frac{1}{3}$. You can see the “correct” answer below.

You didn’t think I had something to say about the waffles as food, did you? I’m sure they are everything one would expect of Eggo waffles. You probably already know whether you consider that a good thing. Tabitha likes them.

P.S. My own father turns 70 today. He certainly supported my own mathematical development growing up. Thanks, Dad! And Happy Birthday.