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.