# 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.

# A book recommendation

Sue VanHattum is a fellow community college teacher and a friend of the project. She cares deeply about math, about parents and kids, and about bringing those three things together for fun and for learning.

She has compiled and edited a book, titled Playing with Math filled with the writing of other wonderful people. Honestly, some of my favorite writers about math and teaching are this compilation. If you don’t have time to seek out amazing writing about math learning on the web, Sue brings Fawn Nguyen, Kate Nowak, Paul Salomon, Malke Rosenfeld, Avery Pickford….so many talented writers and teachers to you in one neat package.

She is crowdfunding the publication of this book. Any contribution helps make the book a physical reality. For 9 bucks, you’ll get an electronic copy. For 25 bucks, you’ll get a hard copy once it is produced. For 5 grand, she’ll come lead a math playtime with your group!

I have put it on my summer reading list. You should too.

Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers has over 30 authors, who each tell their delightful stories of sharing their enthusiasm for math with others. It was lovingly compiled and edited by a teacher whose passion is to share the love of math with as many people as she can.

# [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] 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!

# Patterns on the multiplication machine

Tabitha (6 years old) has been playing with the multiplication machine off and on for a few months now. We have never once used it for learning multiplication facts, but we have had a ton of math-learning fun with it.

Recently, she carefully did an up-down-up-down pattern that she continued along all of the columns.

We discussed her patterning a bit before I sent her upstairs for her bath.

I followed behind and, when I knew the machine was out of reach, asked her a question.

Me: You know how some buttons are up and some are down right now? Are there more ups or more downs?

Tabitha (6 years old): There are the same amount.

This was the answer I expected. As we will see, her reason for the answer surprised me.

Me: How do you know?

T: I did two patterns. Up down up down… and Down up down up. Let me show you.

Me: OK. In this one column [I point to the first column on the left], are there more ups or downs?

T: [She counts each type] Downs. But in this one [the next column to the right] there’s more ups.

Me: So how about the whole thing?

T: Same!

Wait.

I’m gonna figure this out.

1,1,2,2,3,3,4,4,5

There’s more downs.

Me: How do you know?

T: Watch. [She points to columns as she counts, one column for each number word] 1, 1, 2, 2, 3, 3, 4, 4, 5.

Me: Right. But what does that mean?

T: One. There’s more downs. Then one. There’s more ups. Then two, and two, like that. But five is more downs.

Me: Wait. One has more downs, then the other one has more ups. So together the 1s have the same number of ups as downs?

T: Yeah.

Me: So then what?

T: Same for the 2s and 3s and 4s. But 5 has more downs.

Me: So there are more downs total.

T: Yeah.

Me: I have a challenge for you. If you can, try to make a pattern that has the same number of ups as downs. But not now. Now you have to get in the bath.

(to be continued)

## So what do we learn?

We have to keep an open mind when our kids are telling us what they think.

I knew there were more downs than ups because we started with a down in the upper left. Moving across the row, there is an up for each down. The last down at the end of the row has an up at the beginning of the next row.

Every down has an up.

Except for the bottom right corner. That down has no up to pair with. So there are more downs than ups.

This is not at all how Tabitha saw it. She made groups—each column has an extra up or an extra down. Then she grouped these groups—each pair of columns has equal downs and ups.

By keeping an open mind, I was able to listen to her thinking. This let me ask follow up questions, which helped Tabitha make her own thinking better and more clear. I try as much as possible to have explain your answer be about convincing each other of something. You think they’re the same? Convince me. You think there are more downs? Convince me.

## Starting the conversation

Any repeating pattern will do.

Even if the pattern had been Down down up, down down up (i.e. two downs for every up), we would have had a lovely conversation about which there were more of. The keys to the thinking in this pattern were:

1. There were too many things for counting to be a convenient solution, and
2. We started thinking about it when we didn’t have the machine in front of us.

I would have been content if Tabitha had counted 41 downs and 40 ups. But I was very much hoping to push her to use the pattern she had created to reason rather than to count.

# 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.