Addition can grow directly out of children's experience with counting objects. Ben illustrates how a young child adds.
How do children learn to add? One kind of addition grows directly out of enumeration. You can think of enumeration—counting the number of things—as adding by ones. You start first with one and add one to it to get two and then add one more to that to get three and so on. When children figure out how many, they at first count one by one. They count all the members of the first group of things, and then continue to count the remaining things in order to determine the sum. But after counting all for a period of time, the children then sometimes take shortcuts. Suppose a child sees a bunch of things in front of her. She notices that there are three and then two more on the other side of the table. Her solution is to count on from three: “Three… four, five.” So in determining “how many?” children independently discover addition by counting all and then counting on.
Children can work with several different kinds of addition. In this first example, I ask Ben to determine the number of objects in two different groups of chips. He got the wrong number just a moment before, so I remind him to count very carefully.
He gets the right answer, five, for the chips in front of him, and then, without counting, seems to “see” that there are three chips in the other pile. When I ask Ben how many there are altogether, he announces that he has to count them, which he does after pushing the piles together, starting with one and getting the right answer. At the outset then, adding is simply the action of combining two groups (making a union of sets) and most children begin by counting all (starting with one) to get the sum.
Next, Ben gets extremely excited when he hears that we are going to play a new game. He doesn’t tolerate too much chitchat before he asks, “So what do we have to do?” I hide nine pirate coins, one at a time, in the pirate’s secret bag. Then I tell him that I am adding two more. His job is to figure out how many there are altogether.
[This game is modeled on a wonderful task that was created by Zur & Gelman (2004), who in turn based it on a teacher developed activity.]
At first, after whispering the numbers from one to nine, Ben again says, “I give up.” But then just a little scaffolding helps him to get the answer. He first asks what is the number after nine, and when given ten, counts on by one more to get the correct answer. Though at first he proclaimed ignorance, he knows enough to ask for the number after nine. And right after that, like many other children his age, he counted on to get the next and final number. This means that he has a basic concept of addition as counting on from the first addend—at least when he gets a little help from an adult. But didn’t he count all in the 5 + 3 addition-as-putting-together task described above? That’s true, but here the task itself, in which both groups of chips are hidden in the bag, channels his thinking into the idea of counting on. In other words, because he could not see the chips, and would find it extremely difficult to count their images in his mind, he had almost no choice except to count on. So tasks matter: they support or discourage different strategies.
Here’s a slightly different example of Ben’s understanding of addition. This time, I place three chips under a piece of paper, and then two more. He can’t see the result. I ask him to put the same number of chips under his paper. This time, he doesn’t have to tell me the answer; he just has to produce the right number.
The numbers in this task are much smaller than those in the previous pirate secret bag problem. What is remarkable here is not that he easily gets the sum, but that he is aware of how to solve the problem and can describe the needed method in advance. He says, “We have to go three and then two more.” And he proclaims with some joy that the two sets have the “same number.”
Ben and I continued to play various “games” for about 50 minutes. We were getting ready to wrap up when Ben decided it would be fun (he literally jumped up in his seat) to play another game with Janet Eisenband.
She shows Ben another pirate money task, a very difficult one. She chooses eight coins from a large pile of pirate booty. She first gives both of them the same small number of pirate coins, namely, four. Then she hides her four and asks Ben to watch as she takes one of his coins and places it with hers. So now he has three and she has five. I thought she was going to ask how many coins are in Janet’s hiding place. But she posed a much more difficult problem that involved equalizing.
She asked him to select from the large pile enough coins to make his collection have the same number as hers. If Janet’s and Ben’s coins are initially the same number, and if Janet takes away one from Ben and adds that one to her collection, then how many coins does Ben need to restore the initial equivalence? Ben’s response surprised me.
He was able to solve two problems of this type, at least with small numbers. He could have solved the problems by adding everything in his head as he went along. In other words, in the first problem he sees that Janet takes one more for herself, so he adds four to one to get five. He also sees that he lost one, subtracts one from four and gets three. Then he subtracts two from five to find that Janet has two more than he does and that’s the amount needed to make them have the same number once again. Another more general way to solve it is this: Janet took away one from me and gave it to herself. That means that she has one more than she started with and I have one less. So I need to give myself one to make up for the one she took and another one to make up for the extra one that she got. The problem is complex: it requires some adding, some subtracting, some comparison of magnitudes, and some judgment of equivalence.
We cannot tell exactly how Ben solved the problem. At the end, he does not offer an account of his method. But it is clear that solving the problem involves several important ideas and methods, and also requires a great deal of working memory!
Here’s one more example, this time from Ben at the very beginning of his fifth year. We first establish that there are six toy bears under a piece of paper. Then, as he watches, I put two more under the paper and ask him how many there are altogether. Recall that I showed him a very similar problem a year before.
At first, he is quiet, apparently whispering some numbers to himself. Then he pops up and triumphantly says, “eight!” When I asked him how he knew that, Ben says that four and four make eight, and also that he counted on two more from six. I suspect he did that first, and then remembered that four and four is eight. And then he volunteers that, “if you make two more it’s ten.” When I asked him a series of questions involving the addition of two, he got up to 14 and then made a mistake. Despite this, it’s clear that at five years, he is more at ease with addition problems than in earlier clips, seems to understand the equivalence between 6 + 2 and 4 + 4, and even creates and solves a new addition problem that begins with his previous answer.
In brief, these examples show that in a sense, young children already know a good deal about addition before they get to elementary school. Most can figure out what happens when you add by combining two sets and what happens when you start with a set and add more to it. They can deal with some abstract tasks, like doing the pirate’s secret bag task when one set is completely hidden. They can invent and solve problems on their own. They may even be able to describe their own strategies, although many children are less introspective and expressive than Ben, who may be a bit precocious. Finally, the tasks you give a child make a huge difference. Asking the child to combine two visible sets privileges counting one by one, where as the pirate bag task channels thinking into the more efficient counting on strategy.
Why is all this important for you to know? One reason is to understand that children can learn a great deal through their own experience, without direct teaching. You need to give them opportunities to explore and to practice, with no or minimal adult assistance, in a rich environment, with blocks and chips and other things to manipulate, that offers opportunities to engage in meaningful mathematical thinking.
A second reason is to appreciate that children may be more competent in mathematical thinking than we ordinarily imagine. To see this competence you need to give children challenging, carefully designed problems of the kind I have described. If the problems are too easy, children won’t have to think much about them.
A third reason is to realize that our math activities must encourage children’s language, explanation and reflection. In a sense, math education is literacy education. Children need to learn not only how to do such things as label shapes and count out loud but also to talk about mathematical ideas and their own thinking. And they need to see that you are proud of them for doing interesting work.
For additional exercises to accompany this thinking story, visit Using Unedited Videos in Your Courses.