My five-year-old has been able to rattle off the numbers in order into the teens for a while, but when you ask her to count a set of objects, she can only get up to three consistently. “One, two, three–um–five?”
I have an idea of what might be going on in her head. It is pretty clear that she knows that four comes after three. I suspect that she has also figured out that when counting objects you are announcing at each step how many you have pointed to so far. When she points to the first object she knows that she has pointed to one object and announces “one.” Upon pointing to the second object she knows that she has pointed to two objects and announces “two.” When she points to the third things go similarly, but when she gets to the fourth she runs into trouble, because she doesn’t have any good idea how many have been pointed to.
I hear you saying “Whoa, if she knows that four comes after three then of course she knows that after pointing to one more than three things she has pointed to four things.” Not so, because she hasn’t learned that yet. Sure, you know what counting is all about, but kids don’t until they’ve been taught. What my daughter is doing when pointing to the first three objects is subitizing the number of objects pointed to so far. Subitizing is the act of being aware immediately of how many objects are in a collection without counting them. We humans can easily subitize collections of one, two, or three objects.
Up to this point my suspicion is merely that: a suspicion. My hypothesis has to be tested. We need to isolate the pieces of counting. The elements of counting are: 1) reciting the names of the numbers in order; 2) pointing to objects one-by-one; 3) doing 1 and 2 simultaneously; and 4) coordinating actions 1 and 2, saying exactly one number for each object pointed to. We isolate these elements in the case of my daughter by asking ourselves a series of questions:
First, can she recite the numbers aloud, in order consistently? Yes. She has no problem with this up to at least ten.
Second, can she point to objects one-by-one, in sequence? Yes. No problems there.
Third, can she point to objects one-by-one, in sequence while reciting the numbers aloud? Initially we aren’t sure, and we decide that we need to distance this skill from counting itself, so we ask ourselves:
a. Can she recite aloud a sequence of words other than the counting numbers? Sure, she knows the alphabet, as well as many songs.
b. Can she point to objects one-by-one, in sequence while reciting aloud a predefined sequence of words in order? We don’t know initially, but my wife tries it with her and she is successful (for instance, with the alphabet).
Fourth, can she coordinate the two actions above? That is, as she points to objects in sequence and recites words, can she match them up, pointing to exactly one object for each word she says? Again, we aren’t sure initially, but my wife tries it with her and she is in fact able to do it.
This last one is the complete skill of counting. Once she can do that she can count, right? So what’s tripping her up? Well, counting is two separate things: it’s a process as just described, and it’s a way to ascertain the number of objects in a collection. These are distinct facets of counting, but they are intimately connected. You do the process and the last number that you say is the number of objects in the collection. These two facets of counting are usually learned by children separately. It is my understanding that most children first learn the process of counting and only later learn that this process gives you information about how many things are in a collection. A young child who has “learned to count” may count the cookies on a plate “one, two three, four, five, six, seven,” but then when asked “how many are there?” reply with “I don’t know.”
I think that my five-year-old learned things opposite the usual order. I’m pretty sure that she has figured out that counting is all about determining the size of a collection, but initially she is unclear on how the process works. Only by divorcing the counting process from numbers were we able to make any progress. In the last several months she has progressed well, and is on her way to being a proficient counter.
See also: Wikipedia article on counting