Tag Archives: fair sharing

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.

Photo Oct 12, 2 13 41 PM

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.

Chicken skewers

Saturday was a great day. I took Tabitha to dance class, which afforded me an hour to do some reading. Then it was time for lunch and a great family tradition—Free First Saturday at the Walker Art Center.

For lunch, I chose Thai.

We shared an appetizer of chicken satays and Pad Thai. I felt guilty about the lack of adventure in my choices, then remembered that I was teaching my five year old to eat Thai food.

The satays arrived in short order.

Tabitha (five): Oh! Four of them. So we each get two.

I was hungry, so I let this go for a little while. Later, though, I followed up.

Me: You said we would each get two. What if there had been six skewers?

T: I don’t know. I don’t go that high.

T: Four and four. Or maybe three and three.

Me: Three and three. Good. What if there had been three skewers?

T: No answer.

Me: Why ‘no answer’?

T: Because one person gets two and the other person gets one. How is that fair?

Me: Hmmm. Good point. What could we do about that?

T: Split it in half.

Me: OK. Then how much would we each get?

T: I don’t know.

Me: Well, you’d get one whole one and a half. So 1\frac{1}{2} skewers.

T: Right.

Re-reading our campsite conversation from a few months back, I can see that she still isn’t ready to use fractions as numbers. They still don’t really answer how manyquestions in her mind.

After lunch, it was off to the Walker. Which is where we had this fabulous photograph taken. Like I said; Saturday was a great day.

Further adventures in Kindergarten fractions

I took the kids camping this past weekend. Fall along the Mississippi River, mid-70-degree days and 50-degree nights. Pretty much perfect. Having read information about our state park together earlier in the day—including the park’s acreage—Tabitha posed a question.

Tabitha (five and a half): How many acres or miles is our campsite?

Me: It’s only a small fraction of a square mile, but it’s about \frac{1}{8} of an acre.

Tabitha: What’s a fraction?

Me: It’s like when you cut something up. It’s a number bigger than zero, but less than one.

Tabitha: Huh? That doesn’t make any sense!

Me: Well, let’s say you, Griffy and I had three s’mores, and we wanted to share them equally. We would each get one, right?

Tabitha: Yeah.

Me: But what if we only had 2?

Tabitha: Well, then you’d have to cut them in half.

Me: Right. So \frac{1}{2} is a fraction

Tabitha: Oh.

[later]

Tabitha: But what’s the number? You said a fraction was a number bigger than zero, but less than 1.

Me: One-half is more than zero, but less than one.

Tabitha: Half of what?

Me: Half of anything is more than nothing, but less than the whole thing.

Tabitha: But what’s the number? A half isn’t a number!


I have been thinking about the moment when there is a choice to talk math with my kids. I have been trying to understand what I need to know in order to recognize that a choice exists and in order to pursue a mathematical conversation.

Fractions are tough because there really is a lot of specialized knowledge about how people learn them. I have been reading and teaching from the book Extending Children’s Mathematics over the last year or so. The authors make the argument that fair sharing is the best entry point for children’s sense-making about fractions. Not part-whole. Not number line. Fair sharing.

Notice how this plays out in my conversation with Tabitha. I start with part-whole, move to (arguably) number line and she protests that these ideas make no sense.

But as soon as I go with fair sharing, she’s on it. She gets that things sometimes need to be cut up in order to be shared equally.

She also understands—and this is crucial—that halves are meaningless without a referent whole. “Half of what?” is a brilliant and essential question.

So what did I need to know in order to pursue this conversation? I needed to know that there are multiple ways of thinking about fractions, and that fair sharing is going to be helpful for a young child to think about. And that part-whole and number line are going to be dead ends.

Tabitha learns from the conversation that fractions have to do with fair sharing. She doesn’t understand one-eighth—the fraction that initiated the conversation. She doesn’t understand anything more about the size of our campsite, nor about acres, miles or even square miles.

She learns that fractions have to do with sharing. That’s a pretty good Kindergarten-level idea, right there.