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Ethan and the Rocks: One-to-One and Fairness

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This video of four-year-old Ethan illustrates how young children engage in mathematics in real-life situations as he distributes rocks "fairly" using one-to-one correspondence. 

by Herbert P. Ginsburg

One of the basic principles of number is one-to-one correspondence. Suppose that you have some cups and saucers. You put each cup in a saucer—that’s a one-to-one correspondence between cups and saucers. You are sharing the cups equally among the saucers. If there are no cups left over and no saucers without a cup, then the number of cups is the same as the number of saucers. It doesn’t matter if there are three cups and three saucers or three million cups and three million saucers. In both cases, the cups and saucers have the same number. (It’s also true that you wouldn’t know where to put the three million cups sitting in the three million saucers, but that’s not a math problem: you’d need a bigger kitchen.) The serious point is that math is not only about solving problems in the world around us, but is also about ideas.

Does all this sound abstract? Don’t worry. Ethan, age four years and three months at the time of the observation, explains what one-to-one correspondence is all about and how it is related to counting and to justice.

One day when Ethan was visiting his grandfather (me) he rushed into the kitchen saying something, I thought, about rocks. Sensing an interesting mathematical event, I immediately grabbed my iPad and videotaped him (quite clumsily from very odd angles, as you will see). His younger sister was fussing in the background, adding to the normal chaos of children in a small space.

I asked Ethan what he did with the rocks.

He said that he distributed the rocks to everyone. Not only did he say that “Everybody gets one rock,” but also he exuded moral certainty, holding up a finger for emphasis. He was very clear that he didn’t give different numbers of rocks to different people. No, he was insistent that the principle of one-to-one correspondence (not in so many words) be applied.

Analyzing videos of behavior is cognitive detective work: you examine the evidence (what you can see children doing and saying) to discover underlying thinking. Because I didn’t see what Ethan did outside, I had to rely on what he said. This 15-second episode contains many ambiguities. First, I saw enough to know that he did not in fact give rocks to anyone, certainly not to me. So he apparently was talking about a plan, not a reality. Second, what exactly is the plan? If he did intend to use one-to-one correspondence, how did he know how many rocks to get? Perhaps he simply counted out the correct number of rocks. If so, he must have already known the number of people in the house. Was that really what he knew and did? Did he know the number of people in the house? Did he then count out the exact number of rocks so that each person would have one rock?

In the absence of further evidence, we cannot know. It was possible that he incorrectly counted the people in the house and/or the rocks, or just took a bunch of rocks without counting them, or even that he made up the whole story. Detectives need to know what they don’t know, and they must also recognize when the evidence is ambiguous, such that more than one interpretation may be possible. Perhaps the incomplete evidence will make sense later on when more clues will have been obtained. (“Now it makes sense! The butler was the one who put the poison cream cheese on the bagel.”)

So consider what happened next.

Ethan volunteered a short (14-seconds) but complex explanation: “I counted them on the bench out there so I could make sure I have enough rocks for everybody. I did, so I came in and I gave them to everyone.” Again, he really didn’t give them to everyone. He was really talking about what he could do. In any event, Ethan seemed to mean that counting the rocks could solve his problem of giving one and only one rock to everyone. He even stated that his goal was certainty: he wanted to “make sure.” And it’s true that if the number of people is the same as the number of rocks, then each person must receive a rock, and none will be left over.

It’s also important to note that Ethan was able to put his thinking in words, of which he has an unending supply: “counting,” “on the bench,” “out there,” and “enough for everybody.” It’s not easy for little children to be aware of their thinking and even harder to put it into words that others will understand. Mathematics is partly in the individual’s mind, but is also partly a social act requiring clear communication with a community of rational thinkers and speakers.

Next I wanted to find out more about his counting method. So I asked Ethan how many rocks there were altogether.

As he entered the bathroom to wash his hands, he said that he had in fact not counted but only did the one-to-one correspondence. “I only counted—I only said, ‘Julie, Papa, Ethan, Dad, Maya, Mom…” He used a rhythmic cadence to list, one by one, the names of all the people in the house. “I didn’t do regular counting. I did just…”  I started to question him about the last incomplete statement (I was sure that he was doing a one-to-one correspondence) but there was an interruption (a query from the chef about whether scrambled eggs would be suitable).

What’s remarkable here is that he made an explicit distinction between the methods of “regular counting” and one-to-one correspondence. Whether he in fact did what he said is not the point: he was thinking about the difference in methods. Children need to learn to engage in this kind of metacognition, being aware of and able to talk about their different methods of finding solutions.

Next I asked him why he gave everybody one rock.

He explained that he didn’t want anyone to have two because then somebody would not have a rock. The explanation was difficult and he got a little entangled in double negatives but eventually he said, “I didn’t want nobody to have two because then somebody wouldn’t have a rock so that’s why I only give one to everybody.” 

I thought that he was not only concerned with one-to-one correspondence, but perhaps with fairness as well. So I said the following.

I made an interviewing mistake when I started out by saying that the situation he described (two rocks for one person and zero for the other) would be unfair. I should have let Ethan come to that conclusion himself. Perhaps I put words in his mouth and thereby tainted anything he said next. But the certainty and clarity of his response convince me that he was indeed concerned with fairness and could express his concerns very well, even posing a hypothetical situation. “Imagine if I give you two rocks and I had none.” Asked whether he would be upset, Ethan agreed and said, “That’s not right. That’s why I picked one for everybody.”

And then having provided his moral judgment, off he went to watch his favorite television show.

Although the entire episode was only about two minutes long, its content is rich. The main lessons are that in everyday life, a four-year-old understood one-to-one correspondence; he knew the difference between it and explicit enumeration (counting objects); he knew that the two are related to one another; and he could express and justify his thinking in words.

Moreover, he understood all this even though his knowledge of mathematical symbolism was either minimal or nil. This is not at all atypical. Young children do addition without understanding the + or = signs. The same is true of children and adults in non-literate societies. Clearly it is possible to engage in mathematical thinking without any knowledge of the written symbols. By contrast, many students in school use the symbols without understanding the ideas. In a sense, and depending on how math is taught, young children may do more real math before they enter school than after.

We also have learned about the connection between the mathematical and the moral. For Ethan, the idea of one-to-one correspondence was imbedded in the social context and provided a tool for moral judgment and action. Maybe the idea of one-to-one correspondence originated historically in an attempt to regulate social action. We will never know. But it is clear that Ethan used the idea to govern his behavior towards others.

Yet several questions remain. Is Ethan typical? Probably he is not. But my purpose was not to show you the norm but to help you to think about the meaning of one-to-one correspondence, the understanding of it, and the role of language and social context in mathematical activity.

Another question refers to the origins of Ethan’s understanding. How did he learn the concept? It’s really very difficult to tell: the possible influences are many and must interact in complex ways. In everyday life, he may have faced the arguably universal issues of fairness with respect to the distribution of food, as when the parent says, “You have to have the same as Adam. Each of you gets one cookie.” I doubt though whether the parents explicitly described the relation of counting to one-to-one correspondence. Maybe he learned the concept in preschool, through his teacher’s intentional instruction. But research shows that children in cultures without schooling understand concepts like these, so that schooling is clearly not necessary for children’s understanding of some basic concepts. Or perhaps he saw examples of one-to-one correspondence in everyday life, like one foot going in one shoe, and abstracted or constructed the concept from his experience. Or perhaps he learned the concept from the Team Umizoomi television show, which he enjoyed and in fact was going to watch at the end of the episode described here. It’s hard to disentangle all these threads of experience and influence. There are many, many ways in which children encounter, engage, and learn to solve everyday mathematical problems.

early math one to one egg cartonShould Ethan be taught about one-to-one correspondence? He did not seem to need help in talking about it, although many children his age do. One useful approach is to help him formalize his ideas. Soon after the events described, he was helped to produce this egg carton representation. This is a very simple but effective way to structure Ethan’s thought and make it explicit in a visual representation. Each rock goes in a compartment of the egg carton, and in each compartment is the name of the person with whom the rock stands in a one-to-one relationship. Maya gets one, Daddy gets one, and so on. Although the egg carton provides a physical, visual representation, it is also very abstract. There is clearly a rock in each of the five compartments, but there is no person in each. The people are imagined and each is represented by a piece of paper bearing a written name: the egg carton thus teaches a key literacy skill too. The empty compartments suggest that the one-to-one relationship is not limited to these particular elements; more pairs are possible. The use of the egg carton was a good idea that seized on a teachable moment to introduce a first step towards formalization of Ethan’s everyday ideas.

So Ethan may not be typical and we cannot say with certainty how his concept developed. But it is clear that in everyday life children may engage with important mathematical concepts in homely ways, without written symbols, to solve problems such as fairness. And it is clear that we can help children to elaborate and even formalize these basic ideas.

Here is the Ethan video in its entirety.

Several weeks later, Ethan’s sister Maya, who was about one year and ten months at the time, did the following, as reported by her mother:

Maya and I went to a store to buy new sippy cups. I picked a two-pack, in which two sippy cups were enclosed in a practically impenetrable packaging of plastic and cardboard. Maya wanted to hold the package in the car on the way home, so I let her. She then set herself to the task of getting the packaging open, all the while saying, "Ethan, Maya...Ethan, Maya." She probably said this about 20 times over the course of 15 minutes. When she finally got the package open, she proclaimed "Ta-da!" When we arrived home and I went to get her out of the car, she handed me one sippy cup, saying "Ethan." Then she hugged the other to herself, saying "Maya."

So Maya was in stage one of one-to-one correspondence and generosity. One for me and One for you equals two special ones.

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