One thing I found interesting, and pretty useful, on the development of math skills is Stan Dehaene's book
The Number Sense. Dehaene is a developmental psychologist who argues (against Piaget) that kids know a lot more math - and a lot earlier - than you'd think.
The book is interesting in part because of the story it tells about the relation between different number abilities. People have already mentioned the difference between rote counting and 1-1 correspondence. But did you know that even once a child has learned 1-1 correspondence she still needs to learn two extra skills in order to count properly.
Initiating and
completing a counting task both turn out to be totally different skills from keeping track of the number of items once you've started. Thus, there is a certain stage at which it is so common for children to double count the initial and final items (as in "1, 1, 2, 3, 4, 5, 5") that it doesn't even count as a mistake!
BB
