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.

Book shopping

Math teacher mom (and long ago former student of mine), Megan Schmidt sent in the following report for Talking Math with Other People’s Kids month…

Her husband (who is not a math teacher) and three-year-old daughter—we’ll call her Marian—are playing “store”. Marian is trading coins and marbles for books and blankets.

Marian (3 and a half years old): I want to buy a book for mommy to read.

Dad: Pick one and I’ll tell you how much it costs.

M (grabbing a small book from her book shelf): This one is new. Mommy wants to read it to me.

Dad: That one will be 3 silver coins.

Photo Feb 07, 10 06 30 AM

M: 1, 2, 3. Now I want this one (picks a bigger book)

Dad: How much do you think this one should cost?

M: 5 coins!

Dad: How come this one is three (pointing at the small book) and this one is 5? (pointing at the larger book)

M: This book is large, the other is medium.

Megan writes that Marian is quoting Dad here and that Marian’s fondness for the number five may have more to do with her response here than a certainty that five is more than three.

So what do we learn?

Trading stuff is a fun game to play.

You don’t need all the fancy store equipment. A few coins and a few valued objects (here books) and you’re good to go.

There is so much opportunity to mention, discuss and ask about numbers. Fun, fun, fun.

While the idea that 5 is more than 3 is not at all beyond the grasp of a three-year old, I do love Megan’s tentative attitude here. It certainly is possible that Marian considers five more valuable than any number—that the large book should cost five coins because five is the best number, even if the medium book costs 23 coins.

Starting the conversation

A beautiful part of this conversation is when Dad asks Marian, How much do you think this one should cost? 

This question invites Marian to think about and to discuss numbers. It’s lovely, easy to do and is very low risk for both child and parent. It is low risk because there is no wrong answer. Marian is free to set her own price, but thinking about what that price ought to be engages her mind in a deeper way than does simply counting out the coins.

Don’t get me wrong: counting out the coins is a lovely activity too. But How much do you think this one should cost? is a brilliant conversational move that got even more thinking from a three-year old.

How many fives in an hour?

Our local public library has a summer reading incentive program. Children keep track of the amount of time they spend reading, and when they reach 20 hours they get a prize. Some of the prizes are good, including a ticket to the State Fair.

bookawocky_215

To keep track of their time, children get a chart. The chart has 20 individual hours, each represented by an icon. Half of these are circular, suggesting clocks, and half are rectangular, suggesting books. Each icon is broken down into five minute intervals. We were driving home one June Sunday afternoon after picking up Griffin and Tabitha’s summer reading charts.

Me: Griff, each hour on your chart is broken up into 5-minute chunks, right?

Griffin (seven, nearly eight at the time): Yup.

Me: So how many of those chunks are there in an hour?

G: (long pause) Sixteen.

Me: Why sixteen?

G: Well, I thought of 5 minutes like a nickel, and there’s 20 nickels in a dollar.

Me: Wow.

G: So I minused four, because it’s four less.

Me: Right. 60 cents is 4 tens less than 100 cents, though. So I think we need to…

G: (interrupting) Oh! RIght! So…it’s twelve. Twelve fives in an hour.

Me: That’s some really good thinking there, buddy. I wouldn’t have thought to do it that way.

So what do we learn?

If you are new to thinking about people learning math, it may be surprising that asking children to explain their thinking aloud often leads them to correct their mistakes.

Math is very often portrayed as a subject where things are either right or wrong with no in-between. This is not a helpful image of the subject. Indeed, there are many shades between these two extremes. Sixteen was a wrong answer; there are not 16 fives in 60. But underneath that wrong answer is some pretty sophisticated thinking.

When we figure out some new answer based on one we already know, this is called using derived facts. It’s a very useful mental math strategy and it should be encouraged at every opportunity.

You can only encourage it if you know it is being used. And that’s another reason we need to ask about process. We want to know how kids are thinking so that we can help them make that thinking better.

Starting the conversation

While mental math strategies are becoming more explicit in schools, many parents today did not learn many such strategies when they were in school. The emphasis for many parents may have been on (1) memorization of facts, and (2) paper-and-pencil computation. Therefore you may not know very much about derived facts, or more likely, you don’t notice that you use them.

If you have ever thought “58+9 is 67 because 58+10 is 68, and 9 is one less,” you have used derived facts.

Whenever a computation of some kids comes up in daily life, ask your kids to talk through their thought process. Model your own thinking for your kids.

In short, make everyone’s thinking part of the number conversation.

You and they will get better at it as you keep at it.