This is a thinking story about one child’s counting from one to 99 (she says she has to ask her mother about what comes after that). A thinking story is a written narrative presented along with embedded and carefully selected video clips–a kind of multimedia case study–that attempts to bring to life the drama of a child’s mathematical thinking.
A Two Minute Interview with Anna
This is a story with embedded video clips about a girl who is now my favorite counter. At the time of the interview, which took place on an early June near the end of the school year, Anna was a late-four-year-old girl who attended preschool in a low-income neighborhood of New York City. The interviewer was an assistant teacher in her classroom.
The topic of the interview, which lasted less than two minutes, was counting. The goal was to find out how high Anna could count. Why were we interested in this? One reason is that counting is useful. For example, children need to know the counting numbers in order to determine how many, that is, the number of objects in a collection. If the adult says, “You can have eight cookies,” the child can take advantage of the opportunity only if she knows the counting words up to at least eight and then can figure how to take no more, and no less, than eight.
A second reason is that the counting words are systematic, embodying important mathematical principles, particularly organizing numbers by tens. Learning the counting words can involve acquiring important mathematical ideas.
It All Begins with One
The interviewer begins with a simple request, namely to count as high as possible.
The first 10 or so counting words must be memorized in the English language. There is no other way to learn them. Anna knows them all. She begins to use her fingers at four, and continues to use them up through ten, but then does not use them for eleven and twelve.
Many children use their fingers while counting out loud, although they usually start with one. There’s nothing wrong with using the fingers while counting, at least up to ten. After all, our counting system owes a lot to the fingers. The origin of our formal base-ten system, that is, a system that relies on groups of ten, is probably biological: we have ten fingers. Indeed sometimes we call numbers digits, which is also a word for both fingers and toes. Counting on the fingers becomes cumbersome only after ten. (Toes would be the next practical option, but are hard to count unless you are wearing sandals.) Also, finger counting is valuable because it establishes an early link between the counting words and the things to be counted. The fingers help to establish a one‐to-one correspondence between each counting word and each thing.
After ten, Anna proceeds smoothly and accurately to higher numbers.
She gets into a kind of wave, a rhythm, for thirteen and fourteen, and thereafter says the numbers in correct order but seems to run out of steam at twenty-three. How exhausted she sounds! But Anna is an indefatigable mathematician and quickly resumes from twenty-four.
Now she rides her wave all the way up to thirty-nine. But what’s most interesting is her inflection on both twenty-nine and thirty-nine. She stretches out both words and says them like a question before going on. Clearly she sees those words as special. And they are. Each is the last number in a sequence before the next tens-number must be used. In other words, the counting numbers she is trying to say involve a tens-number, such as twenty, followed by the numbers one through nine. When she elongates the twenty-nine, it is as if she is asking, “What’s the tens-number I have to say now?” When she gets thirty, she tacks onto it the numbers from one to nine. That’s the rule: take your tens-number and just append to it all the unit numbers.
After her question-like "thirty-niiiiiiiiiine," she does something remarkable.
We might expect her to ask what number comes after thirty-nine. Or like other children, she might try out thirty-ten, which is not a bad guess: after all, thirty plus ten is forty. Instead, Anna asks what comes after three and answers her own question: “Oh, I know. Four.” She does not actually say, forty, but continues correctly from forty-one to fifty.
Why did she ask what comes after three and how did the answer help her? She seemed to know something about the base-ten structure of the numbers. In other words, she seemed to understand that numbers are grouped in tens: the twenties, the thirties, and the forties. She also seemed to know that the first number in each of tens sequences (twenty, thirty, forty) is related to the unit numbers, two, three, and four. In other words, it’s as if she thought of the tens-numbers as “two-ten," "three-ten," and "four-ten.” If you think of them this way, and you don’t know what comes after the "three-ten numbers," you can easily figure out that next must come the "four-ten" numbers. And from there, it’s easy to see that “four-ten” is very similar to forty. Bingo! It’s interesting to note that the Chinese counting system portrays the tens numbers exactly in this fashion: “two-ten, three-ten… nine-ten.”
So Anna understood some really important base-ten features of the English counting language. She had uncovered its basic structure. This is one reason for teaching counting out loud, without objects. It’s the first case of an abstract mathematical structure that children encounter before school. It has no practical purpose or reward but is interesting in itself. Children greatly enjoy learning to count. There’s no reason why they should not pursue counting as a mathematical puzzle. Children are quite capable of this kind of abstract thinking by age four.
Next, Anna was asked to count higher, to sixty. Probably it would have been better not to give her that tens-number, so that we could learn whether she already knows it or could figure it out. In any event, she continued.
She did her wave and her question-intonation at the end of each tens sequence. She did not know eighty but did ninety correctly. At the end, she thought that her mom might be able to tell her what comes after ninety-nine.
This was quite a performance for a young child. She did many things that are typical for her age group: essentially memorizing the numbers to 20, learning that the unit numbers follow each of the tens-numbers, and pausing before the transition to a new tens-number. Other children make the “thirty-nine, thirty-ten” error and the like. But in my experience, Anna was unique in making the explicit, verbal connection between the unit number and the initial tens-number: four and four-ten.
So that’s the main reason why she is my favorite counter. Another reason, which we tend to overlook, is the sheer joy she had in showing off what she knew and in grappling with the base-ten intellectual puzzle that the counting numbers present.
Finally, watch her entire performance, straight through. The video is just about one minute long but contains a wealth of joyful and meaningful counting. Who knew that this apparently tedious activity could be so interesting?
Here’s a method appropriate for those children who can count out loud to at least 20.
Tell the child that they will learn how to count higher than 20, all the way up to one hundred. Count with the child, “twenty-one, twenty-two… twenty-nine.” Children will easily learn to add the unit numbers to the tens-numbers, and will rattle off the numbers up to twenty-nine. At this point, do what Anna did: ask the child what comes after two. The child will say, three, and will probably not understand why you are asking. But then you can say that three is like thirty, and then ask what comes after thirty. The child will probably easily get to thirty-nine. After that you can ask what comes after three and point out that four is just like forty (why is forty not spelled “fourty”?). Again the child is likely to produce the numbers after the tens-number very easily.
That’s enough for one day. You can repeat as needed and gradually make it up to one hundred. Don’t try to explain much to the child. Just point out the similarity. Remember that children may not catch on after only a few attempts. It may take months. Relax. Use the pedagogy of relaxed patience. Believe that the child will eventually learn the ideas and don’t worry if she doesn’t at first. Be careful not to reward too much. Just acknowledge when the child is right (“That’s right, thirty is like three”), and help her to say the right number if she can’t get it on her own (“No. After twenty-nine comes thirty. It’s just like three.").
For additional exercises to accompany this thinking story, visit Using Unedited Videos in Your Courses.