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Having an almost three year old daughter leads me into deep philosophical questions about mathematics. :-) Really, I am concerned about the concept of "being able to count". People ask me if my daughter can count and I can't avoid giving long answers people were not expecting. Firstly, my daughter is very good in "how many" questions when the things to count are one, two or three, and sometimes gives that kind of information without being asked. But she doesn't really count them, she just "sees" that there are three, two or one of these things and she tells it. Once in a while she does the same in relation to four things, but that's rare. Secondly, she can reproduce the series of the names of numbers from 1 to 12. (Then she jumps to the word for "fourteen" in our language, and that's it.) But I don't think she can count to 12. Thirdly, she is usually very exact in counting to four, five or six, but she makes some surprising mistakes. Yesterday, she was counting the legs of a (plastic) donkey (in natural size), and she had to move around to see all of them: she managed to come to the conclusion that the donkey had six legs. Fourthly, she usually forgets one of the things or counts one of them twice when she is counting to seven, eight or nine. Finally, she never asked her parents what is the number "next" to some other number (say, the numbem "next" to twelve). Now, do you think that she can count? And to how many things can she count?

Having an almost three year old daughter leads me into deep philosophical questions about mathematics. :-) Really, I am concerned about the concept of "being able to count". People ask me if my daughter can count and I can't avoid giving long answers people were not expecting. Firstly, my daughter is very good in "how many" questions when the things to count are one, two or three, and sometimes gives that kind of information without being asked. But she doesn't really count them, she just "sees" that there are three, two or one of these things and she tells it. Once in a while she does the same in relation to four things, but that's rare. Secondly, she can reproduce the series of the names of numbers from 1 to 12. (Then she jumps to the word for "fourteen" in our language, and that's it.) But I don't think she can count to 12. Thirdly, she is usually very exact in counting to four, five or six, but she makes some surprising mistakes. Yesterday, she was counting the legs of a (plastic) donkey (in natural size), and she had to move around to see all of them: she managed to come to the conclusion that the donkey had six legs. Fourthly, she usually forgets one of the things or counts one of them twice when she is counting to seven, eight or nine. Finally, she never asked her parents what is the number "next" to some other number (say, the numbem "next" to twelve). Now, do you think that she can count? And to how many things can she count?

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Rather than answer I will merely invoke a classic Sesame Street episode. Grover is counting oranges: one, two, three etc. And again: one, two, three. Then someone else comes in with a basket of apples and asks him to count these as wells. But he breaks into tears. Alas, he can count oranges but he has never learned to count apples.

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Most of these questions are not so much philosophical as empirical, and there has been a tremendous amount of extremely important work done in the last few decades on children's concepts of number. The locus classicus is

The Child's Understanding of Number, by Rachel Gelman and Randy Galistel, which was originally published in 1978, but this stuff really took off in the late 1990s or so. A lot of people have contributed to this work, but I'll mention two: Susan Carey and Liz Spelke, who are both at Harvard. You will find links to some of their work on their websites. Part of the reason people got interested in these issues is because they are closely related to issues about object recognition and individuation, which had been a focus of a great deal of work just before that. (I.e, people had been interested in the question at what age children start to "pick out" objects from the environment, and to think of them as distinct entities, that continue to exist even when you do not see them. The answers turn out to be every bit as fascinating as one might hope they would be.)It turns out that there are several different cognitive systems at work in numerical cognition. One of them is system that works by "pattern recognition". This is the system that your child is using when she just "sees" how many things there are. She's not counting, even to herself, but just recognizing a pattern. Unsurprisingly, this system does not work for very large numbers. If I remember right, it tends to give out around four or five, for most people---and for many animals, too, who share this particular system with us. There's another system that is based on what people call an "analog accumulator" and that, again, we share with many other animals.

Counting, on the other hand, is something that seems found, in nature, at least, only in humans, though I know there are parrots who have been taught to count. Before we talk about counting, though, we need to distinguish two kinds of counting, which are known as "intransitive" and "transitive". (I think the terms originated with Charles Parsons, but I'm not sure.) Intransitive counting is just rehearsing the number sequence, i.e., saying, "1, 2, 3, 4", etc. Transitive counting is

usingthe number sequence to count some objects. Obviously, you have to learn the former before you can learn the latter, and there is almost always a developmental stage where children are good enough at intransitive counting, but quite bad at transitive counting, or even unwilling to do it entirely.We also need to distinguish the question, "Can so-and-so count transitively?" from the questions (a) how well they do it and (b) whether they understand what they are doing in the way we do. Concerning (a), the distinction we need to make here is Chomsky's distinction between "competence" and "performance". It sounds, from your description, as if your child knows that each item is to be counted (that is, "tagged" with a counting word) once and only once. But knowing this is one thing, and being able to tag each thing just once is another thing. Even we adults make this kind of mistake sometimes, and even a child who almost always makes such a mistake might know what she is supposed to be doing. So in that sense, she might be

ableto count, but not exhibit this ability in her performance.Concerning (b), we adults use counting to find out how many things of a certain sort there are, a fact we then use to make other kinds of decisions. If we count five plates and then go to get forks, we also count out five forks, for the obvious reason that this is one way to make sure we have the same number of forks as plates. But there is, again, almost always a developmental stage at which children can count, but they do not understand the significance of the exercise. So if you ask them to count the plates and then go get five forks, they have no idea what to do. Indeed, at this sort of stage, children almost seem to understand the question, "How many plates are there?" as meaning: Would you please count the plates? You can ask them over and over, and they'll count each time; they won't just think, well, I just counted, so there are five, and why are you asking me again? Nor do they understand, e.g., that, if there are five dolls and five hats, then this means that you have a hat for each doll, but no more, or, conversely, that if you have five dolls and a hat for each doll, but no more, then you have five hats. Amusingly, children at this stage will very often use other "count sequences" instead of number words. They'll count "a, b, c" or "Monday, Tuesday, Wednesday", and be perfectly fine with that. The moral of the story, then, is that the ability to count does not, by itself, imply having an adult understanding of "how many" questions and their answers.

So, does your child know how to count? In some ways, yes, and in some ways, perhaps no.