I interviewed Ben on or very close to his third, fourth, and fifth birthdays. His parents brought him into a room on the Columbia University campus where Ben and I sat at a table to do various "games,” as a videographer captured the interactions. His parents stayed in the room, observing and occasionally interacting with Ben. As you will see, the atmosphere did not involve high-­stakes pressured testing: Ben clearly enjoyed the interactions, which lasted for quite a long time. (After I interviewed him for about 15 minutes, Janet Eisenband interviewed him for another 30. All this with a boy of three years!) His father told me that after the first session that Ben looked forward to playing again with “Dr. Ginsboo."  Here is the story of Ben’s counting.

Spoken and written number words permeate the child’s everyday world and are used for many different purposes. “Here are two cookies for you” clearly talks about how many whereas “Sesame Street is on channel 13” does not. Consider other uses of number such as, “Press the 7 button. We have to go to the top floor.” “You can’t have all three blocks. Give one to your brother.”  “Wow, you weigh 36 pounds!” “We’re going to read the story of the Three Pigs.” “One two, buckle your shoe. Three, four, shut the door.” “How high can you count?” “Two, four, six, eight. Who do we appreciate?” Number words have different kinds of meanings that children must figure out.

Early on, children take a great interest in number words and ideas. Some children love counting as high as they can, like grown-­ups. They may even be interested in the name of the “biggest” number.

At the age of three, Ben had this unsatisfactory conversation with his mother.

Ben: Mama, but what is the LAST number? The one at the end.

Christine: There is no last number. You can count forever without reaching the end.

Ben: Mama, are you sure you're listening to me?!

Let’s try to listen carefully to Ben. Here is part of an interview with him at age three. Janet asks him to check the number of toy bears. Instead of determining how many, Ben seizes on the opportunity to display his knowledge of counting numbers.

So at the age of three, Ben rattled off the number words one to ten, in the correct order. He may have been a bit precocious, but children generally know many number words by this age (Sarneka & Carey, 2008).

A year later, Ben, now four, counts (with his hand in his mouth) up to 17, skipping only 16. After that, with a little help, he makes it up to 20, which he seems to think is the last number.

Note that I helped Ben by starting to count by myself. I did not assume that he knew nothing when at first he said nothing. Also, after he stopped, I did not assume that he lacked the numbers after 17. I asked him what comes after 17, 18, and 19, and he responded correctly up to 20. This kind of scaffolding—gentle hints that help but do not give the answer—can be very useful to learn what a child really knows.

In general, don’t assume that children don’t know something when they fail to respond, or if they give a wrong answer. They might know much more than you think. Your job is to find ways to dig down beneath the surface to uncover the child’s real competence.

We have learned so far that Ben can say the counting numbers up through twenty. But what does he know about them? Watch what happens when Ben plays the catch my mistake game, first after he has just turned three, and then at age four.

At both ages, Ben clearly enjoys turning the tables on an adult, correcting the habitual corrector. The activity is a lot of fun, but also shows that Ben believes several general principles about numbers:

• It is not permitted to skip a number in the sequence. Ben shows that he knows this even at age three when he indicates that something is wrong about saying twenty right after seventeen, although he probably could not fill in the numbers that are missing.
• It is not permitted to say a number twice. At three, he is particularly worried about how the repeated three will multiply, so to speak, in an unmanageable way.
• Numbers are different from colors.

Other children express another important rule: “You have to start with one!”

In brief, Ben, along with other children, knows more about counting than is initially evident. He is not only memorizing the numbers but also thinking about what he is memorizing, even at the beginning of his third year. He has figured out, probably by himself, that the category of numbers is different from the category of colors (would anyone have ever explicitly taught him that idea?), and that there are rules for counting, specifically: don’t repeat a number and don’t skip any. And like other children, he is very likely to know the start-­from‐one rule.

Children’s math, even learning the counting numbers, involves more than memory: independently, young children learn abstract mathematical ideas, even at age three. Parents may be unaware of what their children are learning. Further, as Ben gets older he becomes increasingly able to verbalize his thinking: at age four, he not only recognizes mistakes but also is able to explain why they are mistakes. Language and expression are key aspects of math.

At three, Ben is even practicing to be a cognitive psychologist himself when he decides to play the mistake game with Janet as the experimental subject!

After memorizing the first 12 numbers, learning that number is different from other categories like color, and acquiring rules about skipping and repeating and starting-from-one, English speaking children are faced with a challenge: the words from thirteen to nineteen are difficult to learn. One approach is to memorize them, which would bring to 19 the total number of number words to be learned in this way. That’s a lot to memorize, but there may be little choice. The alternative approach is to learn the underlying pattern, namely that the unit words (one through nine) are followed by teen, which derives from ten. Thirteen is a form of “three-­ten;” fourteen is a form of “four-­ten;” and so on until “nine‐ten.” But the pattern may be hard to detect and the words are strange. "Thir" stands for three, and teen for ten. Fourteen is easy, but "fif" in fifteen stands for five. It makes no sense to teach the pattern because after 19 the pattern reverses, as we shall soon see. A pox on the “yucky teens!”

At age five, Ben has no difficulty with these strange teen words from thirteen to nineteen; he rattles off the numbers from one to 29. 

When children are able to count to 20, they have to learn a new rule for generating numbers. The rule is simple, and underlies our base‐ten system of number: you first have to begin with the appropriate tens-word (twenty, thirty, forty, and so on up to ninety), and then you simply append to it the numbers from one to nine.  So sixty-eight is really “six-ten eight" (which is how it is said in Mandarin, by the way). By contrast, sixteen is really “six one-­ten,” whereas it would make more sense to be “one-ten six” (again, that’s how it is said in Mandarin). The inventor of the yucky teens rule got it backward! In any event, as they begin to learn the numbers above 20, children begin to grapple with the idea that we can think of numbers as groups of tens and units. The idea also underlies our place value system for writing numbers, in which the 3 in 36 refers to three tens and the 6 to six units.

Ben has trouble with the new rule. Instead of thirty, he says fifty and goes on from there, correctly, to fifty‐nine, when he stretches out the nine… as he searches for sixty. As Ben is trying to figure out what number comes after fifty-nine, I pause to let him think. In Ben’s case, the silence was not only golden, but also effective.

After a while, Ben finds sixty and goes on to seventy-­nine, which he says is the last number he can count to. But I’m not so sure. I give him some simple hints.

At first, I just ask him what comes after seven. My intention is to draw a parallel between the unit and tens-numbers. Ben gets it, and then rattles off the numbers to eighty-nine. But there he gets stuck and I need to intervene. At first, I try the indirect method of asking him to think about what comes after eight, but he cannot make the connection between the nine (which of course he knew) and ninety. Given this, I then more or less give him the answer and then throw in one hundred as a bonus.

To summarize, learning to count is not simple, at least in English. Children must memorize the first 12 numbers, and along the way learn the rules about repetition and skipping, and the abstract concept that numbers are a special type of word (as opposed for example to the category of colors). Then children have to learn the yucky teens—the very odd numbers from 13 to 19. And next they have to learn the very sensible base-­ten pattern from 20 to 99 (although it too has some odd words like twenty, which should be "twoty,” and thirty, which should be “threety”). If children have trouble learning to count, send complaints to the inventors of the English counting words, particularly Professor Yucky Teen.