Tag Archives: 7 years old

How many tens?

Here is one from the archives.

Nearly a year ago, Griffin was seven years old and I was doing some thinking about the number course I teach for future elementary teachers. I decided to see how Griffin was thinking about place value.

Me: How many tens are in 32?

Griffin (seven years old at the time): Three, and then two leftover.

Me: How do you know that?

G: Thirty—that’s three tens, and then the zero means no ones.

Me: How many tens in 268?

G: [long thoughtful pause] Twenty-six, and then there would be 8 left over.

Me: What would you say to someone who thought there were six tens in 268?

G: I’d say there are 20 more than that.

That’s my boy.

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If you keep at it, it will pay off…

My son hates “explain your answer” on his homework assignments.

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.

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 sort of book shaped. 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): 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.

Truer words were never spoken.

And if you keep at those little guys, they do come around. I won’t tell him that he just “explained his answer”.

Compare and contrast

Digging deep in the comments to bring out some interesting observations.

Chris Hunter reports the two following interactions with his 7-year old daughter:

First

My daughter comes home from school and tells me she hates math. She completed 13 questions on “The Mad Minute” and compares herself to her best friend who completed all 30.

Second

Bedtime conversation:

Daughter: Dad, did you know that in some countries girls aren’t allowed to go to school? They won’t even get to know what 9 plus 9 is.

Me: Do you know 9 plus 9?

D: No, but I know 10 plus 10 is 20. We do that one a lot.

Me: Okay, so what’s 9 plus 9?

D: 18.

Me: How’d you get that?

D: I counted two down. They make JUST THE GIRLS stay home and do chores ALL DAY.

And Steve Prosser reports a friend’s progress:

third

[She is a] very bright girl, who because she was home schooled never did much math. Finger-counting still in fifth grade. Hated math. Believed she was terrible at it. After just over 1,000 flashes, she “GOT IT”. She was not bad at math, but had never done enough to internalize the basics. Now she loves math (her favorite subject), and is beginning to excel at it.

What century is this?

At breakfast one morning, Griffin (7 years old) asked me,

Griffin: What century do you think this is? The 20th or the 21st?

Me: It’s the 21st century.

G: Oh darn. I was hoping you wouldn’t say that. I’ll have to ask Tabitha and Mommy and a bunch of other people this question.

At this point, I was thinking that he was feeling pretty smart for knowing that the year 2012 is the 21st century, even though there are only 20 sets of 100 in 2012. I was wrong. He was making the opposite argument-we should call the present century the 20th.

G: What about when you were born, in 1970? Was that the 20th century or the 19th century?

Me: It was the 20th century.

G: See, I don’t think that’s right. It should be the 19th century.

Me: Because it starts with a 19?

G: Yeah.

Me: I see. Well, what about the year 50? Not 1950, just the year 50? What century do you think that was?

G: Zero.

Me: Right. Well, we agreed that we would start counting with the first century instead of the zeroth century. So the year 50 was in the first century, and the year 150 would have been the second century.

G: Well, it shouldn’t be that way. I want to start counting at zero. So I’ll keep asking people and find people who agree with me.

A couple minutes later

G: So, have there only been people for 2000 years?

Griffin is an independent, contrarian thinker. If there is a way to think about something differently, or even a way to perform some physical deed differently, he’s all in. A critical thinker in the extreme, this boy never accepts “Because I said so” as an answer. This will serve him well in some areas of life and poorly in others. It makes parenting him a unique challenge.

From a mathematical perspective, it doesn’t matter at all whether Griffin thinks of this as the 20th or 21st century. Sooner or later he’ll give in to convention so that he can communicate with the rest of the population of the first world. The important mathematical thing is to explore the basis and the consequences of the argument he is making.

Is the basis of the argument purely the fact that 2012 begins with 20? Or is there an attention to place value? In other words, is he thinking about 2000 as 20 groups of 100, or just as beginning with 20? My question about the year 50 was intended to get at that. There is no beginning with in this case. He had no trouble, which suggests that he is thinking about groups of hundreds-there are no full groups of 100 in 50, so it should be the zeroth century according to his rule.

This is the consequence of his argument. If you’re going to argue that 2012 is part of the 20th century, you need to be ready to accept the idea of a 0th century.

7 year old math jokes

Griffin is 7 years old. If you have been reading along, you know that we talk math a lot around the house. This is the sort of behavior that results…

On Wednesday of this week, he had a mild spill off his bike as he transitioned from the street to the sidewalk. On Thursday, he made the same transition with ease and grace.

G: Want to know why I don’t fall this time?

Me: Yes

G: Because I did it at the right angle! Get it? The RIGHT angle?

Like I said, there are consequences to talking math with your kids.

This kid’s a real cut up. So one more for your “pleasure”.

Griff was alerting me that his class would be studying two-digit multiplication in the coming week. This will involve the lattice algorithm (little do he and his teacher know just how familiar I am with this algorithm!)

G: If I have any lattice homework, I’ll eat it up! Get it? Lattice? Lettuce?

Not really ready for fractions

Talking Math with Your Kids week continues.

I worry about how much fraction work we do in early elementary grades, before lots of kids are really ready.

Some evidence…

Griffin (seven) has been doing his second-grade math homework, which is to find \frac{1}{4} of 4, and then \frac{1}{4} of 8, and then \frac{1}{4} of 20, etc. After several of these, he is to fill in the blanks: \frac{1}{4} of ___ is ___. He begins with \frac{1}{4} of 4 is 1.

Me: You already did that one; it was the first problem. You have to do a new one.

Griffin (seven years old): But I don’t know what else to do.

Me: Any other number that you haven’t already done is fine.

G: Hmmph.

Me: What about doing 2?

G: I already did that, it’s 8.

Me: No, no. What about doing 2 as the thing you find 1/4 of?

G: Dad, there is no 1/4 of 2. It’d be in the negatives!

I have to believe that some extended time thinking about sharing situations would be a much better use of Griffin’s homework time than expressing the results of this sharing abstractly as \frac{1}{4} of a discrete quantity.

Things that come in pairs

Talking Math with Your Kids week continues.

I was doing the dishes one morning while Tabitha (who was four, nearly five) drew in the other room. She came in for help.

Tabitha: Where are the scissors?

Me: I don’t know. How many pairs of scissors do we have anyway? We have a lot, but we can never find them. Why is that?

T: I just need one pair of scissors.

Me: Isn’t it weird that scissors come in pairs?

T: Yeah.

Me: What else comes in pairs?

T: Pants do. And shoes.

Me: Oooh. Good. What else?

T: Legs. And ears. And noses [giggles].

Me: Noses?!? Noses don’t come in pairs, silly!

T: Eyes do. And glasses.

Me: Nice! Eyes. You know, it’s not just people who have eyes that come in pairs. Fish do, too.

T: Of course!

Me: How many pairs of eyes are in our aquarium?

T: Seven.

Me: So how many eyes is that?

T: Nine?

[Griffin (who was seven years old) wandered in from the living room.]

Me: Griff, how many pairs of eyes are in our aquarium?

G: Seven.

Me: So how many eyes is that?

G: Fourteen.

Me: How did you know that so fast?

G: Seven plus seven is fourteen.

Me: Right. But that’s two sevens. Don’t we need seven twos?

G: Yeah, but it’s the same answer either way.

If kids are going to understand place value, they’ll need to be able to think about different units. Sometimes a unit is a thing (an eye); sometimes a unit is a group (a pair of eyes). Giving them practice counting groups and individual things supports their mathematical development. Helping them notice that some things usually do come in groups supports it too.

Correcting Tabitha when she added 7 and 2 to get 9? That wasn’t nearly so important.

But you’d better believe I talked with Griffin later about things that are commutative.