Division and fractions with a third grader

I found some notes on a conversation I had with Griffin last fall. I do not remember the context for it.

Me: Do you know what 12÷2 is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about 26÷2?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade],  so it was her magic number. We had 13 pairs of kids.

Me: What about 34÷2?

G: Well, 15 plus 15 is 30…so…19

Here we see the role of cognitive load on mental computation. Griffin is splitting up 34 as 30 and 4 and finding pairs to add to each. Formally, he’s using the distributive property: .

He wants to choose and so that .

But by the time he figures out that , he loses track of the fact that  and just adds 4 to 15.

At least, I consider this to be the most likely explanation of his words.

My notes on the conversation only have (back and forth), which indicates that there was some follow-up discussion in which we located and fixed the error. The details are lost to history.

Our conversation continued.

Me: So 12÷2 is 6 because 2×6 is 12. What is 12÷1?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Let’s pause for a moment.

This is what it means to learn mathematics. Mathematical ideas
have multiple interpretations which people encounter as they live their lives. It is (or should be) a major goal of mathematics instruction to help people reconcile these multiple interpretations.

Griffin has so far relied upon three interpretations of division: (1) A division statement is equivalent to a multiplication statement (the fact family interpretation, which is closely related to thinking of division as the inverse of multiplication), (2) Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students—this is the quotative interpretation of division) and (3) Division tells us how many will be in each of a particular number of same-sized groups (Put 12 things into 1 group, and each group has 12 things).

This wasn’t a lesson on multiplication, so I wasn’t too worried about getting Griffin to reconcile these interpretations. Instead, I was curious which (if any) would survive being pushed further.

Me: What is ?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and 12÷6 is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

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Sharing tostadas

It is perhaps not widely known that I love good Mexican food, and that—with the assistance from afar of Rick Bayless—have developed a number of specialties de casa.

Among these specialties is tostadas, which I make starting with corn tortillas. A bit of oil and 10—15 minutes in the oven makes them crispy. We build from there.

The tortillas fit nicely in a 3 by 3 array on my favorite cookie sheet. There are four of us in the family. You can see where this is going, I am sure.

Griffin served himself a second tostada the other night.

Tabitha (six years old): Griffy’s having another one?!?

Me: Yes. There’s a second one for you, too.

T: How many did you make?

Me: Nine.

T: That’s not a fair number!

Me: What would be a fair number?

T: One where everybody can have the same amount.

Me: Right. But how do you know 9 isn’t a fair number? And what would be one?

T: I don’t know.

Griffin (eight years old): Eight would be. Or 40.

Me: Oh! Forty! Then we could each have 10. Would you like to eat 10 tostadas, Tabitha? But then I would need to buy a second pack of tortillas.

T: [Silent, but her eyes get big and she nods vigorously.]

G: Or 20. Or 12.

The final count is 2 tostadas each for Mommy and Tabitha, and tostadas each for Daddy and Griffin. Along the way, I promise Tabitha a taco if she finishes her second tostada and is still hungry. This strikes her as fair.

Rational exponents, third-grade style

I find the tone of this article a bit over the top: It Ain’t No Repeated Addition. In it, Keith Devlin (as is SOP for mathematicians) takes a too-strong epistemological stance–multiplication is not repeated addition.

I am much more interested in the nuanced space between provocative stances. For instance, I am much more interested in a question such as, What is gained and lost in defining multiplication in relation to addition versus some other approach?

Exploring this question allows all knowledgeable parties access to the conversation, and it helps us listen to each other. Telling others that they are wrong tends to shut down the conversation, to discourage listening and make people defensive. Through that lens, I can can read Devlin’s piece in a productive way.

In that spirit, I have engaged with our good friend Michael Pershan on the topic of exponents (By the way; go read this piece—it is excellent.). In particular, I have attempted to ask the analogous question about exponentiation as an operation.

In particular, he has been exploring the conditions under which students confound exponentiation with multiplication. As seen in the very common algebra mistake, .

I have suggested that perhaps the trouble lies in defining exponentiation as repeated multiplication. After a bit of brainstorming, I came up with an alternate definition: doubling (and tripling, etc.)

What if we think of the powers of 2 not as repeated multiplication, but as number of doublings?

This sounds like a trivial difference, and perhaps it will prove to be. But I think it is more than that.

For instance, repeated multiplication makes me think of on its own. But number of doublings suggests to me a starting value (which could be anything) and then we double that value some number of times.

Repeated multiplication doesn’t make clear what to do about , nor . What does it mean to multiply a single 2? Or no 2’s at all?

Number of doublings makes this more clear. means double once, while means do not double your original value.

Rational exponents? Start with mixed numbers and you should be in good shape. One and a half doublings is more than twice what we started with, but less than four times.

What would it be like to start instruction in exponents from the number of doublings perspective instead of from the repeated multiplication perspective?

No better playground for hypothesis testing than a truly blank slate.

Little man knows squat about exponents.

Me: Start with 5 and double it.

Griffin (eight years old): Ten.

Me: Then double it again.

G: 20. Then 40…80…160…2…no…320…640…1280…

Me: Wow.

G: Then two-thousand…five-hundred-sixty.

Me: Holy cow. I did not know you knew that many doublings!

G: Yeah. That’s all I can do, though. I can’t think of what comes next.

Me: Right. Next would be 5120. But that doesn’t matter. We started with 5. Then you doubled one time to get 10. You doubled two times to get 20. You doubled three times to get 40.

G: Yeah.

Me: What if you doubled one and a half times? What do you think that would be?

G: 15

Me: So if you double 5 one and a half times, you would expect it to be 15?

G: Yeah. Is that right? What would it be?

Me: Wait. I want to know what you think here. I will answer all of your questions after you answer a few more of mine. Why do you say 15?

G: Well, doubling once is 10, then half off the next one would be 5 less, so 15.

Me: Instead of 20?

G: Yeah.

Me: What if you had zero doublings?

G: Well…it could be 5. Or maybe 0.

Me: What is the thinking behind 5?

G: You don’t double it at all, so it’s just the same.

Me: And what is the thinking behind 0?

G: Adding and timesing with zero…it’s usually zero. So I think it might be that. But it could be 5. What is it?

Me: I promise I’ll answer all your questions in a minute. One more…What if you doubled half a time?

G: Well…I don’t know….Seven and a half, maybe. I don’t know. I like whole numbers better.

The doublings approach led this third grader to:

1. Linear interpolation for rational exponents, rather than triggering a multiplication schema, and
2. The possibility that (albeit with a low degree of certainty).

These both seem like improvements over the intuitions Michael demonstrates in his piece—intuitions which certainly mesh with the misconceptions with which I am familiar in my middle school and college teaching.

M&Ms

Dessert is a good time to get the children’s attention for a little math talk.

A few weeks back, a smallish serving of M&Ms was about to be given to each child, from a large one-pound bag.

In keeping with my assertion that a day should never pass without asking my kids at least one how many? question, I asked Griffin to choose the size of the serving (but unbeknownst to him that this was the purpose.)

Me: Give me a number between 10 and 20.

Griffin (eight years old): What’s the point?

Me: I won’t tell you until you choose.

G: I won’t until I know why.

Me: Tabitha, pick a number between 10 and 20.

Tabitha (five years old): Twelve.

Me: OK. That’s how many M&Ms you each get for dessert.

G: Oh, then I pick 20.

Me: No. The first number I heard. That’s the one I’m using.

G: You should use the biggest.

Me: Nope. The first.

T: Next time, I should choose….thirteen.

This is beautiful, is it not?

I love the realization that things had not worked out for her maximal benefit. I love that she knows some thinking needs to be applied to the situation.

And I love dearly that the result of this thinking is an increase of a single M&M.

G: No! It’s between 10 and 20!

T: Oh. I should choose…nineteen.

Counting brownies

Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.