Fun with tiles

It is no secret that one of my proudest achievements is creating a lovely space on Twitter where people share stories of children’s math talk. Come read along on the #tmwyk hashtag.

That’s where I came across this tutorial-in-photos.

Math blocks how-to photos

I decided to make myself some. I modified the design a bit (but the food coloring is a genius idea! I used that for sure.)

Then I left them out on Sunday morning and waited for a child to happen along.

Tabitha making a zig-zag pattern with the math blocks

Sure enough, Tabitha began making things.

I ate breakfast in the other room.

Ten minutes later, she came in carrying two tiles, put together so that the blue triangles made a square.

Tabitha (7 years old): A square is just a diamond, but I don’t think all diamonds are squares.

Me: Can you draw me a diamond that isn’t a square?

T: The skinny ones wouldn’t be squares.

Me: Yeah. I think I get it. Draw me one, though.

She proceeded to do so. It took a couple of tries.

I lost the paper, but the result looked something like this.

Skinny diamond

Then, a few moments later she asked a new question.

T: Aren’t all 4-sided things squares?

Me: The doorway isn’t. One of those tiles has four sides but isn’t a square.

I  quickly draw a parallelogram in my notebook.

Non-rectangular parallelogram

Me: This isn’t

I drew another 4-sided shape.

Concave quadrilateral

Me: This isn’t either.

T: That has 3 corners, not 4. So it can’t be a square.

Me: Show me the three corners.

She counted the three corners that point out from the center of the shape, missing the one that points back inward. She paused.

T: Oh…four.

So What Do We Learn?

Opportunity to think about math is important. Something as simple as leaving an interesting math object out for children to play with can lead to fun math talk.

Tabitha was working on the definitions of square and diamond in this conversation, and she was paying attention to the properties of shapes. This is important work for elementary children. When children are very young—before about first grade—they are learning to identify shapes based on appearances. As they move further into elementary school, they need to start paying attention to properties—the number of sides, the number of vertices (“corners”), etc.

Starting the conversation

Make some of these tiles. The materials cost me less than $20 (mostly for the wood—I probably could have gotten it a lot cheaper), and the dying and painting took about an hour on a Saturday evening. Then leave them out.

Or leave out a bunch of squares, triangles and rectangles you cut out of construction paper (you can do this for under $3 and less than 10 minutes of cutting).

Then let the children play and be ready to talk.

 

Math in the alphabet

The children attended a well-run chess day camp this summer. Good people running things; a warm and welcoming atmosphere. Lots of varied activities to keep kids’ bodies engaged as well as their minds.

Sadly, this takes place on the complete opposite end of the Metro area from where we live. We had to drive all the way across St Paul, Minneapolis and deep into St Louis Park during rush hour. Ugh.

This led, one day, to my trying to find a topic of conversation to keep at least one of the children occupied while we drove home. I recount for you this conversation below.

Me: Tabitha. Can I ask you a question?

Tabitha (7 years old): Sure.

Me: What letter comes before I in the alphabet?

T: H. That was kind of an easy question.

I love that she has turned into a critic. If I am not challenging her, she calls me on it.

What she has not seemed to notice yet is that these questions she deems easy are just my openers for the good stuff.

Me: Yeah. Here’s a harder one. What letter comes two letters before S?

There is a fairly long pause here. This is a harder question because of how most of us know the alphabet—forwards. If we want to know what is 2 less than 71, it is not so hard to count backwards. We have lots of experience counting backwards. But we don’t have so much experience saying the alphabet backwards, so we need to make up a strategy.

T: Q and R.

Letter squares for q, r and s

Me: Q is two letters before S, yes. Now you ask me one.

T: What letter comes after Z?

Brilliant. What a great question. I wish I had thought of it myself.

Letter squares for w, x, y, z and three blank squares

Me: Oooooo. Good one. I say A. I say it starts over.

T: Nope.

Griffin has been listening in but not participating. He sees his chance to get in on the action.

Griffin (9 years old): Negative A.

Me: Wouldn’t that be what comes before A?

G: No. It comes after Z. It’s negative A.

T: Nope. Not that either.

Me: OK, then. I am stumped.

T: Nothing.

Me: Huh?

T: Nothing. No letter comes after Z.

So what do we learn?

This is a more sophisticated version of another mathy letters conversation I had with Tabitha a while back. Back then, we were trying to figure out which of two letters comes first in the alphabet. Here, we are more paying careful attention to precise placement (two letters before, not just before).

The other interesting thing going on is our three different ideas about what comes after the end.

My idea: After the end, we go back to the beginning, like the days of the week.

Tabitha’s idea: There is nothing after the end. It just ends.

Griffin’s idea: The end is like zero. When you get to the end, you repeat what you already had, only using negatives.

It is OK that we didn’t resolve who is right.

Starting the conversation

About a year ago, I started making a habit of having the kids ask me the next question. I highly recommend it.

You know how your children are always testing the limits of rules in everyday life? Like you say, “Do not touch” and they see how close they can get their finger to the forbidden object without actually touching it? That is normal and necessary behavior on the part of children.

They will do it in the world of ideas, too. Tabitha did not choose “What letter comes after Z” at random. She chose it because she knew it would be interesting to talk about. It probably would not have occurred to me to ask it. Our conversation was richer because she did.

 

Pistachios

My father buys things in bulk. Not the bulk bin, dispense-a-little-bit-into-a-plastic-bag bulk. Costco bulk. Sam’s Club bulk.

The children and I spent some time with my father and stepmother (who are wonderful, loving people) at the Wisconsin Dells recently. We shared a rented condo. They brought bulk snacks.

Did you know that you can buy graham crackers in a container that holds four of the usual boxes of graham crackers?

What need does one family have with FOUR BOXES of graham crackers?

More to the point, they brought pistachios. I forget to check whether it was a three-pound bag or a four-pound bag but it was an awfully large bag of pistachios.

The image below is a small fraction of the total.

Sandwich bag stuffed full of pistachios in their shells.

While we were in the condo, Tabitha (7 years old) took her first interest in pistachios. Her brother Griffin (nearly 10 years old) has been a fiend for them for years. One day, Tabitha announced something to me.

Tabitha (7 years old): I threw out eight pistachio shells.

Me: And what do you learn from that?

T: I ate four pistachios.

Me: How do you know that?

T: Four plus four is eight.

Me: Nice. And five plus five?

T: Ten!

We carried on this vein for a little bit before we got distracted.

A couple days later, I was rushing around preparing for a work trip. Tabitha was again snacking on pistachios.

T: Is 13 an even number?

Me: No. Why do you want to know?

T: I must have counted my pistachio shells wrong. I must have missed one. So it’s 14.

Me: And what does that mean in terms of pistachios?

T: I ate 12. No. That can’t be right.

Me: Oh! I think I know how you got 12!

At this point, I was headed downstairs to get something to put in my suitcase. By the time I got back up, both of our minds were on to different things.

We never did get to a solution, nor did I find out how she got her wrong answer.

So what do we learn?

Tabitha is playing around with the every pistachio has two shells relationship. She is thinking about ratios: Two shells for every one pistachio.

A child does not need to have mastered multiplication, or fractions, or division to think about these things. I have written about ratio thinking from young children before. Ratios come naturally from repeating a process. Eating a pistachio produces two empty shells every time. Sharing candy produces one piece of candy every time. And so on.

Starting the conversation

In light of this, help your child notice for every relationships. There are four wheels for every car. There are four legs for every chair. There are two wings for every bird. Point these relationships out and have your child do the same. Consider the exceptions (have you ever seen a 3-legged chair?) Count up how many wheels there are on two cars, and on three cars.

Eat pistachios.

Postscript

I have two theories about her answer of 12 pistachios for 14 shells.

1. She tried to figure it out by thinking about 10 and 4. Half of 4 is 2. She added that back to the 10 and forgot that she still needed to find half of 10.

2. She subtracted 2 from 14.

I like theory 1 a LOT better than theory 2 because it matches the ways she has been thinking so far. Using subtraction seems unlikely when she knows this is a different sort of problem.

But of course I do not know for sure.

Nights of camping

The following conversation took place in the run-up to our annual summer camping trip recently.

Rachel has no interest in camping, so this ritual is all mine. I started the little ones young with a one-night trip within an hour from home so that we could come home if it’s a total disaster. As they have aged and we have developed our routines, we have gone further afield, exploring wide-ranging Minnesota state parks for two-night stays. We added a weekend fall trip, too.

Last summer, the kids began to ask why “we only go for two nights”.

Ladies and gentlemen, when the kids ask that question, you know you’re doing it right.

So this summer we are expanding to three nights. Tabitha was thinking about that change the other day.

I am straightening some things on the front porch, sweeping and tidying. Not thinking about anything in particular.

Tabitha (7 years old): If we’re going for three nights, is that 2 days and 2 half-days?

Me: Yes.

A few seconds pass.

I realize that I have an opportunity here.

Me: How did you think about that?

T: Every night is a day, except the last one, when we go home.

Me: What if we went for a whole week’s worth of nights? What if we went camping for 7 nights?

T: Easy. Six days.

Me: And?

T: Two half-days.

Me: OK. Ready for a hard one?

T: Yeah!

Me: There are 365 days in a year. So what if we went camping for 365 nights?

T: [slowly] Three…hundred…sixty…four!

Me: Nice!

T: I can even do 400.

Me: You mean 400 nights of camping? You know how many days that would be?

T: Yeah.

Me: All right. Tell me.

She does.

Later, she is in the shower. I am not-so-closely supervising nearby. I get an idea.

Me: Tabitha, what if we wanted seven days of camping?

T: How many nights?

Me: Right.

T: Eight. Am I right?

Me: I can’t trick you at all, can I?

T: Ask me another!

Again, a sign that things are going well. Contrast with her claim a couple years back, “Sometimes I don’t want to tell you about numbers because it’s just going to turn into a big Daddy math talk!”

I have to think hard to dig up something that will be more challenging for her.

Me: You want a hard one? A really hard one?

T: Yes!

Me: Last year, we went camping twice. Altogether, we camped 4 nights. How many days did we have?

T: Three…five…

It turns out that Griffin is lingering in hallway outside the bathroom. He chimes in.

Griffin (9 years old): Four.

Me: Two days, and four half-days.

G: Right. That’s four.

Me: But she’s thinking about it as four half-days, since they aren’t attached to each other. I can see an argument either way.

This summer’s trip was to Lake of the Woods in the far northern reaches of Minnesota.

Griffin posing with an oversized walleye statue in Baudette, MN

So what do we learn?

It may surprise some readers that I have filed this conversation under Algebra.

Like many of the other algebra posts, we are not using x or y, or making graphs or solving for variables. Instead we are thinking about a relationship, and about what that relationship looks like for a wide variety of numbers.

The relationship we are working with here is a simple one: one less. Tabitha had noticed that the number of full days we camp is one less than the number of nights we camp. She had even generalized the idea—notice that she didn’t count the days individually. She said, “Every night is a day, except the last one.” This answer doesn’t depend on any particular number of days; it works for all numbers of days.

What I did in this conversation was help her to apply this idea. By asking her about a wide range of numbers of days, she got to feel the power of her generalization. That is algebra.

The other important part here was continuing the conversation while she showered. Thinking in reverse is an important mathematical skill. We had started with how many days do we get with a certain number of nights? I moved us to how many nights do we need for a certain number of days? The fancy math word for the relationship between these two questions is inverse.

Starting the conversation

Camping trips, vacations, trips to grandma’s house…these are all opportunities to have the conversation we had. If your child doesn’t ask about it, you can ask your child. We are going to grandma’s house for three nights—how many days will you have to play with your cousins while we’re there?

More generally, there are two Talking Math with Your Kids moves I want to emphasize.

  1. It took me a moment to notice that Tabitha had offered me an opening for conversation. I was thinking about something else at the time. When I noticed it, I put those other thoughts aside to talk, ask and listen. That part of the conversation took probably 2 minutes. We can all spare 2 minutes to get our kids’ minds working. We just need to notice the opportunities.
  2. I followed up later on. Following up is good for two reasons: It lets you and your child examine an idea more deeply, and it helps cement memory of the conversation. We remember something we revisit multiple times better than something we only think about once.

Hints at Holiday

I told an abbreviated version of the following story on my math-teacher blog, where I used it to drive home a point to my colleagues. This version is for parents.

My wife had been out of town for several days. I was tired of doing all the cooking and dishes. It was a lovely Saturday evening at the end of a busy day.

It was time for nutrition lessons.

It was time to get dinner at Holiday.

Sign in front of gas station advertising milk prices

Oh right, like you have never done this.

The constraint was this: The kids had to select something from each of the four major food groups (do not try to talk to me about that new food pyramid; I will not listen.) They needed a meat/protein, a fruit/vegetable, a dairy and a grain.

Griffin (9 years old): Do donuts count as a grain? They have a lot of flour in them.

Me: Scratch that. WHOLE grain. No. Donuts do not count as a grain.

It turns out that the whole grains are hardest to find.

Tortilla chip and bag with picture of chip.

At Holiday, you’re not going to do much better than tortilla chips, whole-grain wise.

As a mathy bonus, Griffin later noticed that the claim underneath the picture of the chip on the bag reads, Enlarged to show texture and detail, but that the image is the same size as the chip.

But back to our story.

Tabitha (7 years old) had brought along money to buy some hot Cheetos.

She was under the impression that they would cost $1.35, and she had her money ready. Five quarters, one dime. She even had me check that these coins totaled $1.35.

When she got to the front of the line, it turned out that they Cheetos cost $1.49.

It would have been fun to talk about the difference in price here, and have her fish out the right amount to make up the difference. But there were people in line behind us. We needed to move this along.

I told her to get two more dimes out of her coin purse and give them to the man. I intercepted the change so as not to give away the answer to the question I was about to ask, and we turned to leave.

Me: You owed him 14 more cents and gave him 20. How much change should you have gotten back?

Tabitha seemed confused by my question. It was not that she was unable to answer it; rather she did not understand the whole getting change thing. I made a mental note of this and pressed on.

Me: You gave him 20 cents when you only owed him 14 cents. So you get some money back. How much should that be?

Still nothing. It seemed the money/change/debt thing was getting in the way of thinking through this number relationship. So I switched tactics.

By this time, we are outside, strolling slowly home.

Me: How much more is 20 than 14?

This question put her in a different frame of mind. She slowed down and looked dreamily into space. She was thinking.

Tabitha (7 years old): Thirty-four? or maybe thirty-five?

Ugh. Right answer, wrong relationship. I think she cued in on the more in that sentence.

I tried one last time to trigger the thinking I know she can do.

Me: Let’s try this. Fourteen plus something is 20. What is the something?

There was a long, thoughtful pause.

Griffin interrupted the pause.

Griffin (9 years old): How old were you last year?

T: Six!

Me: Did you work that out, or did you say it because Griffy said it?

T: Griffy.

Griffin and I had talked about this before. But we talked about it again on the way home—about how it is important for Tabitha to have the opportunity to think things through for herself. I tried to anticipate his needs: (1) to demonstrate that he knows, and (2) to help his sister.

If he needs to demonstrate that he knows, he can:

  • Say he knows but keep the answer to himself,
  • Write it down,
  • Ask if he can whisper it in my ear.

If he honestly wants to help his sister, he can ask a question that will help her think. How old were you last year? does not help her to think about the relationship between 14 and 20. But How much more is twenty than fifteen? might help her think, because she has often counted by fives.

So what do we learn?

We learn that it is sometimes quite difficult to get the right question that will get a child to think. Context, time pressures, level of difficulty, mood, the presence of siblings…all of these things can conspire to cut off the thinking.

But if you are persistent in the moment, you may get somewhere.

And if you are doing this every day, you’ll eventually hit the sweet spot.

Most of all, we learn that it is the thinking that matters, not getting the kid to say the right answer.

Starting the conversation

Persistence is key. I didn’t get where I wanted in this conversation. You won’t get there sometimes either. That’s OK.

Ask your question, adjust it if necessary. Let it go if you need to.

There’s always another day.

What makes a sandwich

The 3-year old daughter of fellow Minnesotan, fellow math teacher and friend Megan Schmidt made the following proclamation a couple weeks back.

This simple claim has led to lots of fun conversation. Let’s call the daughter veganmathpup (since she is the daughter of Twitter’s @Veganmathbeagle), or VMP for short. 

All discussions with VMP are filtered through her mom via Twitter. All discussions with my own children are my best recollections of the recent silliness.

Open faced sandwiches

Veganmathpup’s assertion boils down to this: A sandwich needs these things: (1) a slice of bread, (2) a filling, (3) another slice of bread. I wanted to know about open-faced sandwiches. Is an open-faced sandwich properly called a sandwich? VMP was silent on this matter. So I asked Tabitha.

Tabitha (7 years old): That counts as a half-sandwich…actually more than a half-sandwich.

So an open face sandwich is not actually a sandwich for Tabitha. This gave me a chance to introduce the term misnomer.

Cookies

A week or so later, VMP claimed that “2, 3, 4 or 5 cookies can make a sandwich”. This was a clear violation of the earlier rule here. Two cookies, no filling? How can this be a sandwich when “It takes three things to make a sandwich”?

So I asked about Oreos. Does VMP think of an Oreo as 1 cookie? 2 cookies? Most importantly, Is an Oreo a sandwich? Megan related the following conversation.

Megan: [Handing VMP an Oreo]  VMP, I have a question.  Is this a sandwich?

VMP (3 years old):  [Examining carefully] Um, no.  It’s not.

Me:  Why isn’t this a sandwich?

VMP:  It doesn’t have things, like a burger.

Me: [Handing her two Oreos stacked on top of one another] Is this a sandwich?

VMP:  [Examining even closer this time] No.  it doesn’t have stuff in it. It needs lots of stuff inside like a burger to be a sandwich.  I want a burger.  Let’s get one [face full of oreos]. We won’t tell Daddy.

So many follow up questions I was unable to ask here. Does a Double Stuf Oreo have enough stuff inside to count as a sandwich? What about a Mega Stuf Oreo? Close up of a Mega-Stuf Oreo.

A Mega Stuf Oreo contains approximately 3.1 times the Stuf of a regular Oreo.

Marshmallows

Then the plot thickened.

Megan went on to report that, even after opening the Oreo to demonstrate that there is a filling, VMP rejected the Oreo as a sandwich because the filling is white.

Allow me to summarize:

  • Three things are required for a sandwich.
  • Unless they are cookies, in which case you only need two.
  • An Oreo is one cookie, so is not a sandwich.
  • Even if you want to call the Oreo wafers cookies and the Stuf the filling an Oreo is still not a sandwich because the filling is white.
  • The filling in a sandwich is properly referred to as a burger.

I saw a flaw in the logic, though.

Three marshmallows: mini, regular and giant

Marshmallows are white.

I asked about this. Megan reported that their marshmallows are colored.

I HAVE BEEN FOILED BY A THREE YEAR OLD!

So what do we learn?

Children have ideas.

Children use their minds. They think about things.

We can contribute greatly to our children’s learning by probing those ideas.

Formulating precise definitions is an important part of doing mathematics. Sorting things into examples and non-examples is part of this process. It really doesn’t matter whether we are sorting shapes (square, not square) or food (sandwich, not sandwich). And when the child is three years old, it really doesn’t matter whether she is consistent in her sorting.

What matters is that she is thinking in this mathematical way.

Starting the conversation

You can do as I did. Tell your child that another child says it takes three things to make a sandwich. Ask your child whether she agrees. Then ask about open face sandwiches and about Oreos.

But the bigger picture is important here too. There is a useful habit to develop as a parent—ask follow up questions when your child makes proclamations.

Other conversations we have had in this vein include Spirals, Circles and Armholes.

Counting grapes

I am pretty sure I have mentioned this before, but one of my proudest achievements has been watching a “Talking Math with Your Kids” hashtag (#tmwyk) blossom on Twitter in the past few months. Now, on a nearly daily basis I (and you, if you join us over there) get to see conversational gems such as Kindergarten kids talking about Spirals and cool math prompts such as Counting Grapes.

Michael Fenton—a father and math teacher—sent this photograph into the #tmwyk world recently. Naturally, I had to talk with Tabitha and Griffin about it.

Two bowls—one with five grapes, one with eight half-grapes

The conversation with Tabitha (7 years old), I captured on video.

Here’s the transcript:

Me: Which one of these bowls has more grapes?

Tabitha: (7 years old): [points to a bowl, probably the one on the right but hard to tell] Obviously!

Me: What do you mean, ‘obviously’?

T: I mean look at this! One, two, three, four, do you mean halfs?

There is a thoughtful pause.

T: Actually…

She points to the bowl on the left.

T: Cause these are halves

Me: But how do you know that there’s more here than here?

T: Cause look.

She uses her thumb and finger to indicate that halves of grapes are getting put into pairs to make whole grapes.

T: One, two, three, four

Now she shifts to the bowl on the left and counts the whole grapes individually.

T: One, two, three, four, five.

So what do we learn?

The key moment is right here: I mean look at this! One, two, three, four, do you mean halfs? (This occurs 8 seconds into the video.)

That is when she notices—on her own—that half grapes are not worth the same as whole grapes. It is where she shifts her attention from items (of which there are 5 on the left and 8 on the right) to whole grapes (5 on the left, but only 4 on the right).

The rest is tidying up details. The learning happens in that one brief moment of insight.

Starting the conversation

Ask your own child this question when you have a spare moment. Don’t correct or interrupt. Just listen. Object if their explanations are incomplete, but otherwise just listen.

Technical notes (and acknowledgements and thanks)

This was our first video using Google Glass.

There will be many more, I am sure. I’ll write more about this in the future, and I am happy to discuss with any interested parties. (You can hit me through the About/Contact link here on the blog.)

In the meantime, I want to thank Go Kart Labs for their sponsorship and financial support. They funded most of the cost of my Google Glass through a generous donation. These folks are smart, kind and interested in the overall goal of the Talking Math with Your Kids project, which is developing a world full of intelligent, creative and curious citizens. Upstanding people who do beautiful web-design work here in Minnesota.

The equal sign

It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems.

But I am tired. Wiped out and exhausted.

So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?)

This plan worked beautifully for about five minutes.

She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook.

Eight plus four equals box plus five

Me: What goes in the box?

Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is…

Hey! This doesn’t make any sense!

Me: Why not?

T: 8 plus 4 is something, then plus 5?

Me: What does the equal sign mean?

T: Is. Like 2 plus 2 is 4.

Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1?

T: No!

I let it go and we move on with our evening.

Later on, though, after putting on jammies but before toothbrushing, I follow up.

Me: Tabitha, I want to ask you a follow up math question.

T: OK.

Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1?

T: Yes! Of course! Easy!

Me: Can I let you in on a little secret?

T: A secret secret? Or not really a secret?

Me: Not really a secret. But something you might not know.

T: [rolls eyes] OK.

Me: The equal sign means “is the same as”.

T: Of course! I know that!

Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1.

T: Oh.

So what do we learn?

This is kind of a big deal.

We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!)

But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as.

Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right?

Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do.

The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right.

You can follow Tabitha’s and my adventures in equality in the coming weeks.

Starting the conversation

If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun!

Postscript

Coincidentally, a fourth-grade teacher wrote up his class’s explorations in equality today. If you’re interested in what this can look like in school (easily adaptable for homeschool), head on over.

Tens again

Slowing down at the end of a long, active spring day. Stormy clouds are rolling in. Tabitha and I watch them together for a couple of minutes from the west-facing window at the top of our stairs.

I ask Tabitha if I can ask her a quick math question.

She consents.

Me: How many tens are in 32?

Tabitha (7 years old): Three.

Me: So quick! How do you know that?

T: 10, 20, 30. Easy.

A silent moment elapses.

T: And there’s 10 tens in a hundred.

Me: Yes. Lovely. So true.

How many tens are in 200, though?

T: Twenty.

Me: Whoa!

T: Yeah.

Another silent moment elapses.

T: Asking “How many tens are in 30?” is like asking “How many ones are in 2?”

Me: Wow. I had never thought of it like that. And is it also like asking “How many hundreds in 300?”

T: Except I don’t know that one.

Me: You don’t know how many hundreds are in 300?

T: No.

Me: Three.

T: Oh. I thought it was tens in 300.

So what do we learn?

The power of silence and of conversations in quiet moments. Both times a silent moment elapsed in this conversation, Tabitha continued with an idea of her own. And both are gems.

And there’s 10 tens in a hundred. Many grade school worksheets have attested that there are 0 tens in 100, when what they really mean is that there is a 0 in the tens place in 100. We can do a lot more mathematics with the ten tens in 100 conception than we can with the 0 tens in 100 one.

Asking “How many tens are in 30?” is like asking “How many ones are in 2?” This right here is powerful stuff. For Tabitha, ten is such an important part of the structure of numbers that it behaves like one. Ten, for Tabitha, is a unit—a thing that you count.

If you are new to this blog (and many of you are—Welcome!), you may not have spent four minutes with video. Do so now, please. Consider it your Talking Math with Your Kids homework. It’ll be fun. Promise.

Starting the conversation

Wait for a quiet moment. Ask for consent. Ask How many tens are in 32? Listen, follow up and allow a few moments of silence.

Twister

Tabitha received a Twister game for her recent birthday (7 years old!) She enjoys a version of the game in which one person spins and the other follows instructions until, as Tabitha puts it with much delight, the cookie crumbles.

The players switch roles for the next round. No score is kept.

She wants to play a round one recent Sunday evening. I have been writing, so I have her set it up in the kitchen while I finish up.

She comes back to me with questions.

Tabitha (7 years old): Daddy! What’s six plus six plus six plus six?

Me: Wait. How many sixes?

T: Four.

Me: Twenty-four.

T: Yes! I counted them right!

Me: Huh?

She takes me into the kitchen to show me the Twister board.

twister

T: See? One, two, three, four, five, six.

She is counting the green dots in one row.

T: Then one, two, three, four

She is counting the rows.

Me: So four sixes is 24. Nice. Can I show you something cool? It’s also six fours. See? One, two, three, four.

I am counting the dots in one column—each a different color.

Me: Then one, two, three, four, five, six.

I am counting the columns.

Me: So four sixes and six fours are the same.

T: Like the dominoes.

She is referring to a recent homework assignment in which dominoes were used to demonstrate that 6+4 is the same as 4+6, and that this is true as a general principle about addition.

So what do we learn?

Rows and columns are fun, fun, fun.

Malke Rosenfeld of Math in Your Feet reminds me regularly that children love to play in structured space. She uses blue tape on the floor for her math/dance lessons and has noticed that children love to play freely in and around the spaces created by the tape (seriously: click that link, have a read and then go buy some painter’s tape!). The same thing is true for the Twister board. It creates a structured space for Tabitha to explore at a scale that allows her to use her whole body. That’s a good time for a seven-year-old.

But children don’t always notice the rows and columns in an arrangement like the Twister board. They need to learn to notice it. This is an important step on the path to learning multiplication. The fact that our conversation began with “What is 6+6+6+6 ?” tells me that Tabitha notices the rows and the columns. She knows that the answer to 6+6+6+6 should be the same as her count. By introducing the language of “four sixes” and “six fours”, I am trying to help her notice the multiplication structure underlying her ideas.

Starting the conversation

Arrange things in rows and columns. When you do, the whole thing is called an array.

Point out arrays in the world. Count the number in each row together, and count the number of rows. Notice together whether the numbers switch if you count the number in each column and count the columns. Does eight rows of six become six columns of eight? Does this happen for all numbers?

Here are some of my favorite arrays.