Children’s experiences with partitioning [TDI 3]

If you watch this video, you will see a pretty standard U.S. treatment of introductory fraction material.

PLEASE understand that this is not about Sal Khan or Khan Academy. What you see in that video is what happens in many, many elementary classrooms across the U.S. on any given day. It is what is written into our textbooks (pre-Common Core, of course—we’ll get to Common Core in week 6).

I did not have to work very hard to find additional examples to support my claim. Here is a tutorial on Sophia. Here is something from “Your destination for math education”. And here is a self-paced math tutorial at Syracuse.

I am not cherry picking straw men here.

To be clear, we introduce fractions with a part-whole model. A circle (or rectangle) represents the whole. We cut the circle (or rectangle) into some number of equal-sized pieces—that number is the denominator. We shade some of those pieces—that number is the numerator. That I am pointing this out surely makes some Decimal Institute attendees uncomfortable because how could it be any different? I’ll get to that in a moment. Stick with me here as I build a case pertaining to decimals.

If you believe that defining an abstract mathematical object and then operating on that object is the most powerful way to teach mathematics, then there is no logical objection to starting fraction instruction with decimals.

After all, children know something about our base-10 place value system by the time they get to third grade. They know something about the decimal point notation by then, too, as the result of money and (sometimes) measurement. (Oh, and calculators—don’t forget the calculators.)

So why not put all of that together and have tenths—the very heart of the territory to the right of the decimal point—be the first fractions they study? If you believe that children learn mathematics as a logical system that is little influenced by their everyday experience then there is no reason not to.

From a logical perspective, halves and tenths are the same sorts of objects. Tenths come along with a handy notation and so—from a logical perspective—are simpler than halves.

Indeed, it is much much easier to train children to get correct answers to decimal addition problems than it is to train them to get correct answers to fraction addition problems—even when the fraction addition problems have common denominators. (Sorry, no research link on this. Ask your nearest upper elementary or middle school teacher whether I am talking nonsense here.)

But we cannot fool ourselves into believing that ease of obtaining correct results has any correlation with grasping underlying concepts. Children can be trained to give correct answers without having any idea what the symbols they are operating on represent.

Take the video linked here, for example. (In it, I do a Khan Academy exercise using a purposely flawed way of thinking and score approximately 90%—I get an A without showing that I know anything useful.)

This leads us to the work of the Cognitively Guided Instruction (or CGI) research project from the University of Wisconsin. This project studied the ideas about addition and subtraction situations, and strategies for working them out, that children have before formal instruction begins.

It turns out that they know a lot.

Of particular importance is their finding that when teachers know how students are likely to think about addition and subtraction problems, and when teachers know the strategies students are likely to use, these teachers are more effective at teaching addition and subtraction.

In short, CGI demonstrated—for addition and subtraction—that better understanding the cognitive structure of addition and subtraction makes you a more effective teacher.

In the years since that first set of results, the team has extended their results to initial fraction ideas. In the book Extending Children’s Mathematics, they argue that the cognitive way into fractions with children is fair sharing.

That is, the ideas that children bring to school prior to formal instruction having to do with fractions are those that come from sharing things. Sharing cookies, cupcakes, couches and pears; children have cut or broken these things in half, considered whether the resulting pieces are equal in size, and decided whether the sharing is fair many times before they study fractions in school.

When you do start with fair sharing, children’s ideas about how to do this follow a predictable path. Halving and halving again are common early ideas even when sharing among three or five people. Similarly, children share incompletely early on. When they need to share one cookie among 3 people, they will suggest cutting into 4 pieces and saving the fourth for later.

This more recent CGI research demonstrates that paying careful simultaneous attention to (1) the number of things being shared, and (2) the number of people doing the sharing is a late-developing and sophisticated skill that comes as an end product of instruction.

You can see this in a conversation I had with my children over the weekend (written up in full on Talking Math with Your Kids).

In that conversation, we had 2 pears to share among 3 of us (real pears, not textbook pears). Griffin (9 years old) suggested cutting them into thirds, but then got distracted by the campfire before correctly naming the amount we would each get. Tabitha (6 years old) worked with me to half and half again. Only once we had a single remaining piece right there in front of us did she suggest cutting that piece into 3 pieces.

The concrete conversation created a need for thirds. But thirds only occurred to her once that need existed. As long as we had whole pears or halves of pears, we could keep cutting in half.

Here was the end result of that sharing.

Now back to decimals.

The CGI fraction work constitutes persuasive evidence that not all fractions are cognitively equivalent. While starting the study of fractions with tenths makes sense from a logical perspective, CGI demonstrates that children do not learn from logical first principles.

They learn by considering their experience.

Children have lots of experience with halves. We might expect thirds to be just as obvious to children as halves are, but it isn’t true.

So let’s take seriously the idea that experience in the world has an effect on how children learn. And let’s accept that this fact should have an effect on curriculum design.

Then if you still want to teach decimals before fractions, you would have a responsibility to demonstrate that children have anywhere near the real-world experience with tenths that they do with halves and thirds.

When we discussed on Twitter recently children’s real-world experience with tenths, we came up with:

• money (where the connection to fractions is weak, see also week 2’s discussion on Canvas),
• pizzas (about which I am skeptical, see next week’s interlude),
• metric measurements, and
• not much else.

In comparison to the tremendous amount of work children have done with halves and halves of halves (and halves of those), how can tenths be the first fraction they study in school?

Summary

To summarize, I am arguing:

• That part-whole fraction work makes logical sense to experienced fraction learners,
• That children do not learn fractions by logical progression from definitions, but by connecting to their experiences with situations in which fractions arise in their everyday lives,
• That we have research evidence for this latter claim,
• That the truth of this claim should have implications for how we teach decimals to children, since their experiences with tenths are much less robust than their experiences with simpler fractions, and that chief among these implications is…
• That we ought to reserve serious decimal work until kids have developed the major fraction ideas about partitioning, repartitioning and naming the units that result.

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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.

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.

Task design

For the last two years, I have been working with the four surviving original authors of Connected Mathematics on a revision that is responsive to the Common Core State Standards. My task has been top-to-bottom revision of three of the four rational number units, Bits and Pieces I, Bits and Pieces II and Comparing and Scaling.

The process for this work is unusual for commercially available U.S. curriculum materials, so I want to share a few observations from the inside. They will trickle out over the next few months, and they’ll get filed under “Curriculum”.

The task

In CMP2, we focused the initial fractions unit on careful introduction of the number line. The premise was that children had lots of elementary school experience with area models for fractions, and that we wanted to introduce the more sophisticated linear models.

We introduced the licorice lace problem.

In this problem, a group of four kids is going on a hike. They have a 48-inch licorice lace and they want to share it equally among themselves. Sid (the protagonist in our narrative) carefully marks the places where he will make the cuts.

Just before he actually cuts, two more kids show up. Now they need to make new marks on the already marked-up lace. The cycle is repeated a couple of times. At each phase, we ask students to name the part of a licorice lace each hiker receives.

If you try this yourself, you will notice that it’s pretty hard to locate the marks for sixths when there are already fourths marked. Not impossible, but hard.

Ideally, some students in class will try, and some students will go to twelfths; others will go to twenty-fourths. Then when it’s time to name the fractions we have sixths, twelfths and twenty-fourths on the table and we can talk about equivalence and partitioning linear things.

From classroom feedback and my own experience working the problem with adults (both in professional development and college courses), it was clear that the problem needed a redesign. The set up was wordy, using one and a half pages of text to work through a small set of tasks. The marks before cutting were slightly implausible. The sharing and re-sharing was too complex for simple problem-posing.

The redesign

Two years ago, I took on the task of redesigning this problem.

I knew we needed something that was (1) linear, (2) shareable, and (3) already marked.

Linear and shareable are properties of licorice laces (these are not Twizzlers, each of which-while shareable-is not plausibly shareable among four children). That third criterion was new. If I could find something whose pieces were already marked, I could get rid of the complicated storyline and a tremendous amount of text.

“Marked pieces” is important because this is a problem about partitioning and repartitioning. We want kids to have pieces that they need to cut up further, and to have to think about names for these new pieces.

Skittles would not work.

These, while delicious, are mathematically unproductive for our purposes.

A bag of Skittles is composed of the original unit, one Skittle.

I needed something where we partition the original unit. It is perhaps shameful how many hours of thought went into this. But I eventually found it.

It’s perfect. Linear, shareable and already marked. You want to share equally? Each person gets , sure. But can’t you see that each gets ? Or that it could also be ?

It is helpful that most people don’t know the standard partitioning of a Tootsie Roll. (Did you know it was ninths? Be honest, now. It hasn’t always been; it used to be sevenths.) If you don’t know the standard partitioning of a Tootsie Roll, then we can make as many pieces as we like to start each new task. No more marking and re-marking. We just give a new  Tootsie Roll and a new number of people.

We know from research that sharing is a productive context for understanding fractions. We’re sharing something that is already partitioned, so we need to repartition when the number of sharers is not a factor of the number of pieces.

Feedback from classrooms and my experience working with adults (again-professional development and college courses) suggests that we get more mathematics with a lot less effort setting up than we did with the previous version.

Summary

Not every problem in Connected Math has gotten this level of attention, of course. But a lot of them have. This is a curriculum that takes context seriously as a basis for mathematical activity and abstraction.

Once we have committed to a particular mathematical development (e.g. partitioning in linear situations in order to move to the number line), we seek a problem in which the right mathematical activity naturally results. I am proud to have been a part of that.

There are unshareable sizes?

Enough about me. Let’s get back to overthinking our teaching, shall we?

This came in the mail today.

Seems innocuous enough. No goofy combination discounts. No bait and switch. Just a straight-up buy one, get one free offer.

But wait. Buy one what?

One “shareable-sized flatbread pizza”.

Contrary to what we have been teaching kids in middle school classrooms, Cosi is suggesting that there are sizes too small to share.

There are units of which we cannot take a fraction.

The rational numbers are not dense.

Who knew?