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.

Dots!

The New York Times published an article about Common Core homework this week.

As is going to be the case with a news article (in contrast to say, a post on a blog dedicated to children’s mathematical ideas), one can’t really learn any mathematics from the piece. One critique got hit twice, though—that children are being forced to draw lots of dots.

Here near the beginning of the piece:

Ms. Nelams said she did not recognize the approaches her children, ages 7 to 10, were being asked to use on math work sheets. They were frustrated by the pictures, dots and sheer number of steps needed to solve some problems.

And a bit later:

Her daughter, Anna Grace, 9, said she grew frustrated “having to draw all those little tiny dots.”

“Sometimes I had to draw 42 or 32 little dots, sometimes more.”

I have no interest in picking up political issues surrounding the Common Core State Standards on this blog.

But I do think a parent frustrated by all those dots deserves an explanation of what all those dots are for.

Before we begin, please be assured that there is absolutely no mention of dots in the Common Core. What is mentioned is the array. An array is a collection of things arranged in rows and columns. We have discussed arrays before here at Talking Math with Your Kids. They are very useful tools for representing an important meaning of multiplication—that multiplication is about some number of same sized-groups.

Arrays (with dots or other things) are useful tools for making these groups visible, either actually visible or visible in the mind.

So I asked Tabitha (7 years old) to draw some dots for me.

Me: Tabitha, [neighbor girl and best friend] wants to play. Before you go outside, can you draw that picture for me? Three rows of five dots.

Tabitha (7 years old): That’s easy! Fifteen.

She is probably counting by fives here. She completes her picture for me.

Array of dots: 3 rows of 5.

I know that neighbor girl is waiting. I decide to press my luck.

Me: What if it had been 3 rows of 6?

There is a long, thoughtful pause.

T: Eighteen!

Me: How did you know that?

She shrugs her shoulders. Now is not the time to force things. Neighbor girl is waiting. So I offer a strategy.

Me: Let me tell you how I think you might know it.

T: OK.

Me: Six is one more than five. So each row would have an extra dot. That’s 15 for the 3 rows of 5, and then 16, 17, 18.

T: [smiles] Yeah.

We share a high five and she is out the door for a morning of clubhouse shenanigans in the backyard.

Quick note: Tabitha does not let me get away with stating her strategies incorrectly. I have done this before—summarized how I think she is thinking—and when I get it wrong, she objects. I am glad about this.

So what do we learn?

This is what those dots are for. They give us something we can talk about. Without those rows and columns, the conversation is so much more abstract. We were picturing those dots in our minds as we talked about counting them.

The three rows of five she drew gave us a jumping off point for imagining the three rows of six we discussed. Three groups of five now has a relationship for her to three groups of six.

More importantly, the strategy of finding new facts based on old facts (here that 3 groups of 6 is 18 based on knowing that 3 groups of 5 is 15), has been introduced explicitly. It is something we will talk about in the future, and something she will know to consider.

Without the array, it is not at all clear to me that she would have been able to know what 3 groups of 6 is. She could have drawn 3 unorganized groups of 6, I suppose, and counted them individually. But this is a much less sophisticated strategy, and she is ready for more than counting individual objects.

Starting the conversation

Many children do not naturally see rows and columns. Given an array, they may haphazardly count the objects around the edge, then in the middle. This often leads to double counting and skipping things.

But even children who are very good at keeping track of their haphazard counting—and who can get correct counts every time—may not see the row and column structure of an array.

So put 15 pennies in 3 rows of 5. Have your child count them and notice whether she counts in rows and columns, or whether she counts in some less structured way. Model the counting yourself so that she can see an example of the rows and columns at work. Don’t worry if she doesn’t see the structure yet, but do make a note to do more of this kind of counting in the future—seeing the structure of an array is an important stepping stone to multiplication and to the measurement of area and perimeter.

Then have your child put things in rows and columns.

Or just have her draw dots.

 

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.

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.

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.

10-minute reading time

A while back, bedtime was spiraling out of control. The kids share a room; they would be wound up at bedtime and the transition to sleep was not happening smoothly. We had a big, big problem on our hands.

We solved the problem with 10-minute reading time. The kids have to be in their beds. We dim the lights. We set a timer for 10 minutes. It has to be quiet during that time. Then we turn out the lights, give them something to picture in their minds, and sleep comes more easily.

Complete transformation. It is awesome.

One night, Tabitha (5 years old at the time) wanted to color. We talked and agreed that she could do it “sometimes”. As is the nature of 5-year olds, she soon wanted to know the limits.

The following conversation took place on a Wednesday night.

Tabitha (five years old): I know I can’t color every night but can I tonight?

Me: Yes.

T: Then read, then color the next night?

Me: I don’t know. I think reading twice before the next color is better.

To be clear. It was not my intention to get into a math conversation at this point. I just wanted her to go to bed (Warning! Link Not Safe for Work, and Possibly Offensive to Sensitive Ears. But Funny).

No, this move on my part was truly about literacy, not math. I don’t want 10-minute reading time to turn into 10-minute coloring time. I really, really like the idea that books will become part of my kids’ independent bedtime routines.

But Tabitha loves to know the rules she’s playing by. And when those rules are based on numbers, they’re going to lead to math every time.

T: So read-read-color-read-read-color…like that?

Me: Right. That sounds like a good ratio.

T: Or read-read-read-read-color-color?

Whoa.

Couple things.

First of all, I used the term ratio with absolutely no expectation that she would process it, and I am quite sure that she did not. I have long been an advocate of using good vocabulary with my children—there is no shame in not knowing the meaning of a word, but also no sheltering them from the fact that these words exist. This is at least partly the source of their substantial vocabularies. But I do not believe she knows the word ratio.

Secondly, Tabitha’s reformulation of the 2:1 ratio as 4:2 blew me away. It nearly slipped past me without notice. I was focused on getting them to bed; we were in the truly final phase of that process. I had pretty much tuned her out.

But when I looked at her, I could see she was expecting a reply. She needed to know whether she could get two coloring nights in a row by doing four reading nights in a row.

So I replayed her question in my mind, counting the reads.

Me: Yes. That would be fine. You may do that.

So what do we learn?

Together with the recent Easter Candy conversation, this makes clear that young children are thinking about early ratio ideas.

Think about this for a moment. What is the same about these two sets of tiles?

tile.ratio.2

And what is the same about these two sets of tiles?

tile.ratio.3

In each cases, it is the ratio.

In the first case, there is one blue for every yellow up top and also down below. This is what L was working on when she offered a second chocolate egg to Tabitha.

In the second case, there are two blues for every yellow. This is what Tabitha was working on when she asked about read-read-read-read-color-color.

Traditionally we think about ratio as a sophisticated fractions topic that needs to wait for early adolescence. I certainly would not want to be held accountable for teaching 5-year olds ratio and the associated notation. But their everyday experiences allow for them to think about these ideas. As parents, we can keep an eye out for those opportunities and talk about them when they arise.

Starting the conversation

Both of these ratio conversations with 5-years olds have resulted from constraints. Five year olds love to test rules. “Yes” and “No” are inflexible and allow no wiggle room. This is sometimes desirable. Yes, you must leave now to get to school on timeNo you may not leave the house without pants. Yes you must look both ways before you cross the street.

But when we are guiding our children’s behavior, we would be wise to allow some space for the little ones to test the boundaries. Does it really matter—in the big picture—whether L eats one chocolate egg or two? Does it really matter whether Tabitha colors two nights in a row instead of reading? I say no. Neither of these things really matters.

What matters is that L does not continuously gorge on candy, and that Tabitha has some alone time with books. Constraints rather than absolute mandates seem to have encouraged mathematical thinking in both of these cases while also addressing the big picture.