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.

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Griffin on thirds

This one is from the deep, deep archives.

When Griffin was five years old, he drew the picture below.

Griffin (five years old): I’ll make the boiler one-third red, one-third green and another third red.

Me: Where did you learn about thirds?

G: You told me.

Me: When?

G: I don’t remember.

Griffin seems to have absorbed a part-whole model for thirds that roughly matches how young children often think of halves.

Compare to his performance two years later. In that later conversation, Griffin was struggling to think about one-fourth of two. This pretty much matches a major argument in the book Extending Children’s Mathematics by a subset of the CGI research team. Their claim is that a part-whole fraction model is not very useful as the beginning place for thinking about fractions; that instead the extension of sharing division to complete and equal sharing involving remainders is a more natural place for beginning fraction instruction. See also Tootsie Rolls.

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.

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.