The idea of how many is deep, abstract, and cognitively demanding, involving learning number words and figuring out of the number of objects in a collection. Ben illustrates learning what how many really means.
Three-year‐old Ben could rattle off the numbers from one to 10. He had the numbers memorized, but he did not know how to use them to find out exactly how many toy bears were in front of him. Saying the counting numbers in order is different from accurately counting things. I call the first counting (saying the number words, or counting out loud), and the second enumeration (finding out how many). You can count without being able to enumerate but you cannot enumerate without knowing how to count.
The Idea of How Many
To an adult, the idea of how many seems simple enough. There are four bears here or 10 candies there. But for young children, enumeration is not easy. They must learn methods for figuring out the number of objects in a collection, and they must also learn what how many really means. Indeed, the idea of how many is deep, abstract and cognitively demanding.
Suppose we have the collection of objects pictured here. We use words to name each. From left to right, we call them lion, school bus, and perhaps icky. If you refer to the school bus as lion, people will think that you need some serious help.
Suppose now that we count the objects. We point to the lion and say “one,” to the school bus and say “two,” and to the icky and say “three.” And then we decide to count them again. This time we point to icky first and say “one,” next point to school bus and say “two,” and finally point to lion and say “three.” If we count from right to left, the lion is three. And if we count starting in the middle, the lion can be two. The number words are clearly not like names. Names designate specific objects (Harry the lion) or categories of objects (lions), but we should not use number words as names. Neither one nor three is the name of the lion.
This can be very confusing for young children who are used to words referring to things, and are constantly trying to learn new words. One of my students asked a child to count. She said, “one, two, three, five.” When asked why four was missing, the child replied that she was four. She seemed to use the counting word as a name for herself.
So one of the first difficulties in enumeration is that counting words are used differently from ordinary names. We can refer to the animal as lion but not tiger or football. But we can refer to it as one, two, and three (or indeed any counting number). This violates ordinary usage and confuses young children (and sometimes adults too, as I will show later).
Why this odd usage? What do the counting words refer to? To understand the issue, consider the basic ideas that underlie counting.
We must say the number words in the proper order. One cannot come after four. By contrast, the order of counting things does not matter. You can start with icky or you can start with lion.
We have to count each object once and only once. If we use the word one to refer to the lion, we can’t use it to refer to the icky as well. I once interviewed a child who, when asked to count some chips, began by arranging them in a circle and then counted the objects over and over again as he looped around the circle (which is the worst way to arrange objects if you want to count them). I had to stop him in order to continue the interview.
We can count any discrete unit: dogs, imaginary unicorns, or even more imaginary ideas (“I had two ideas today”).
Not only that, we can count any combination of things. A group can include two dogs, 52 unicorns and one idea—or anything else.
Physical arrangement is irrelevant for counting. You can count the objects when they are scattered around or when they are in a line.
The physical nature of the objects does not matter. Every object, no matter what it is, is a single unit for the purpose of counting. This idea must be very strange to little children. Suppose you find that one group has three frogs and the other three horses. Frogs are small and horses are large. How can both groups be three?
And finally, here is the really big and difficult idea: each number in the sequence does not refer to an object but to the number of objects (the cardinal value) up to that point. Adult counting is so automatic that we sometimes don’t appreciate the complex ideas behind it. But if we are to teach effectively, we must try to see the world from the child’s point of view. In other words, we must overcome our adult egocentrism.
Imagine that you see these pictures of a strange looking arctic creature in an exotic land called Riverside Park. You want to find out the number of creatures in total: the basic issue of how many.
You begin by drawing a line enclosing the first picture and say “one” because there is one creature. Next you draw a line around the first two and say “two” for the same reason: there are two creatures within. But when you point to the second object and say “two,” that object itself is not two. The word refers to the numerical value: the cardinal value of the set, the collection of two arctic monsters. Then you draw a line around all creatures and say “three.” Again, three refers to the whole group, not to the third creature. One way of thinking about this is that you start with a set of one, and then that set of one is included within the larger set of two, and then that set of two (which contains one) is included in the larger set of three.
I’ll bet that many readers have not thought about counting things in this way, as sets within sets, always increasing by one. The idea is not simple and it’s therefore no surprise that children have trouble with it. Let’s see how Ben deals with how many.
Ben Enumerates (Figures Out "How Many")
At the outset of the first interview, at age three, Ben displays variable command of enumeration.
At first, given six “elephants,” he counts to five. Not bad: only one off. When the elephants are turned into “bananas,” he again counts to five, very quickly, not bothering to count each with care. He spews out the number words very quickly, sometimes, as in the case of the yellow chips, not even looking at what he is pointing to. Yet, he was close, only off by one, on the first two problems: that’s pretty good for a child turning three. He is also inconsistent. He gets three wrong, calling it two but then when I take away one, he says two again, and does not appear to be troubled by the inconsistency. And then, at the end of the sequence he succeeds in counting three elephants and four elephants, as he points to one at a time very carefully.
Later, about 25 minutes into the interview (an impressive amount of time for a three-year-old) the interviewer, Janet Eisenband, gives Ben a slightly different enumeration task, asking him to produce a certain number of bears.
Ben has no problem with producing one and two bears (although he is concerned to select specific bears, perhaps because he has not mastered the rule that a number can refer to any object whatsoever).
But when asked to produce four, he puts out two and then three and finally four (although he referred to the last collection as "four"). Asked to check the last result, his counting goes berserk, as he spews out the number words from one to ten.
Now let’s turn to Ben at age four.
At first, Ben finds it easy to produce the small numbers three and five. Each time, he carefully slides one “banana” towards me and arranges the result in a straight line, carefully setting it apart from the remaining chips. Certainly during the past 12 months, his production has become more deliberate and controlled.
Then we turn to a bigger problem, namely eight chips.
We see that he gets a little confused after he sets aside six. After all, producing larger numbers requires more working memory (mental space in which to carry out and monitor mental operations on pieces of information) than does production of a smaller number. As children get older, their working memory expands, as does their repertoire of strategies for dealing with the task literally at hand.
When I suggest that Ben check his result, he notices that the chips appear to resemble the numeral 4. That of course does not help him solve the problem, but it shows that he has numerals on his mind. After that, he tries again to put out eight, but this time is a little sloppy about where he puts the chips, failing to push them aside or place them in a straight line so that he would count each chip once and only once. The result is that he gets an incorrect answer. Further, it’s not even clear that he is really finished at the end. His inflection indicates uncertainty, and I may have stopped him too soon.
In brief, Ben does well with a relatively large number, but needs to perfect strategies for careful enumeration. One lesson for educators is that children may get wrong answers for many reasons, one of which is that they are sloppy and lack strategies that can guard against carelessness and overcome the limits of working memory. A wrong answer, therefore, doesn’t necessarily mean that the child doesn’t understand the basic concept of counting, just as a right answer doesn’t necessarily mean that the child has mastered counting.
The next important issue is whether Ben understands cardinality, the idea that the last number in the count sequence signifies the total amount. You might expect that, at age three, Ben might not understand it. To find out, I introduce an elephant cardinality problem.
Ben does not seem to understand cardinality, and each time has to recount the elephants to determine their number, even when it is only two. Also, it’s interesting that as he counts the four elephants at the outset, Ben says something like “one, it’s a two, it’s a three, it’s a four.” This is very close to saying that each number is the name of that particular elephant with which it is paired.
What about Ben at four years? He was certainly able to enumerate larger numbers than he could at age three. To learn about his understanding of cardinality, I covered up the chips that he had just counted and ask how many are under the sheet of paper.
After Ben says “zero,” apparently referring to the number of chips on top of the paper, I remind him again that he had previously said that there were eight chips. But he was overwhelmed, and in a gesture of defeat, spread his arms and said, “I give up.”
Next I use Piaget’s classic conservation of number problem: will Ben see that the number remains the same even if you simply rearrange the objects into a new and different looking array? This task would seem to be easier than the hiding problem because Ben can now see everything that happens to the objects: nothing is hidden. I asked him to give me five apples and he did.
In order to determine the number of apples after I rearranged them, Ben had to count them all over again, just as he did when they were hidden. He did not seem to know that the last number counted indicates how many, and that moving the objects around does not change that number.
In brief, learning enumeration is complex and difficult. It requires a new use of words to refer to concepts of number (like “fiveness”), not to things. It involves basic and deep mathematical ideas, such as the irrelevance of physical attributes for number: three little things are “more” than two huge things, even though the latter are larger than the former. Children take a long time to master the ideas and the strategies required for enumeration, such as carefully counting each thing once and only once so as to reduce the demands on memory.
Two Lessons Learned
What general lessons can we take from Ben’s story? One is to appreciate that early math is not simple: young children need to grapple with important mathematical ideas and develop strategies for determining how many. We saw that at ages three and four, Ben did not fully understand that the number of objects does not change when they are hidden, and that by age four, Ben had developed a strategy to make sure that he counted every object once and only once: he moved objects aside as he counted them.
A second lesson is that a correct answer does not necessarily indicate understanding. Sometimes Ben counted correctly but failed to understand that the last number counted indicates the total amount. A right answer can be a smokescreen for ignorance. As is well known, this phenomenon extends from preschool to graduate school, and well beyond!
How Can We Teach "How Many"?
Children love stories. Stories often have a lot of math embedded in them, as in the case of the three bears. The bears range in size from small (baby bear), to medium (mama bear), and to large (the somewhat overweight papa bear). The bowls of porridge are small, medium, and large. The baby bear gets the small bowl, mama bear gets the medium, and papa bear gets the large and downs it in one gulp. You can say that the bears are one variable (small, medium, large) and that the bowls are a second variable also (small, medium, large). The variables are in a one-to-one relationship in which small bear is paired with small bowl, middle bear with middle bowl, and large bear with large bowl. This is a kind of simple function. Children could not understand the three bears story unless they had some comprehension of relative size (small, medium, large) and simple correlations (one-to-one correspondence, as when small goes with small, medium with medium, and big with big).
You can use stories to teach children how many. Click here to check out an interactive storybook, the Monster Music Factory, which focuses on enumeration.
You can also use planned classroom activities to teach how many. Here is one set of organized lessons from Building Blocks (Clements & Sarama, 2007):
Students begin by practicing how to make the numeral 4. This in itself is not very interesting, but children do have to learn to write the numerals, and the Building Blocks curriculum wisely introduces written numbers in conjunction with the corresponding number of objects. Thus, the children learn to write "4" in the context of counting four. Notice too that the curriculum introduces counting the different parts of the written numerals (three straight lines, two of which are shorter) and their relative positions (horizontal and vertical). Many things we do and encounter in everyday life, including written numerals and letters, can be “mathematized,” or conceptualized in explicit mathematical terms.
After this, children work with small numbers of objects on plates. They engage in the activity of determining which plate has more objects, and also in determining how many are in each set. The curriculum encourages subitizing, that is, seeing the number of the set without counting. Children should be able to glance at a group of objects and know right away that there are two or three or even four. Of course, if children cannot subitize a set, they are helped to count to determine its numerical value.
Finally, pairs children engage in a get just enough activity in which they try to find a set that matches the number of another. They might start with four pencils, and then try to find, somewhere else in the classroom, some other group that also has four objects, or they might take four objects (for example, some blocks) from a larger collection. This activity gives the children a meaningful task to accomplish and encourages them to use counting to solve the problem (that is, to determine whether the new set is the same number or more or less than the four pencils).
How many words are there in this last sentence?
Clements, D. H., & Sarama, J. (2007). SRA Real Math Building Blocks Teacher's Resource Guide PreK. Columbus, OH: SRA/McGraw-‐Hill.
Huang, Y., Spelke, E. & Snedeker, J. (2010) When is four far more than three. Psychological Science, 17 401-‐406.
Sarnecka, B. & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662-‐674.
Sarneka, B. & Gelman, S. (2004). Six does not just mean a lot: Preschoolers see number words as specific. Cognition, 92, 329-‐352.
For additional exercises to accompany this thinking story, visit Using Unedited Videos in Your Courses.