The studies presented in this book will be of interest to anybody concerned with the teaching of arithmetic to young children or with cognitive development in general. The book provides an extremely detailed account of the different types of counting behavior of half a dozen children over two years. The "teaching experiment" used investigates children's construction of counting schemes, writing operations and their systems, lexical and syntactic meanings of number words and, finally, thinking strategies. The data allowed the authors to reach their main goal: to document the many subtle changes in children's counting and to interpret them theoretically. At the same time, the results of their intensive study lead the authors to affirm that a major shift in the arithmetic curriculum is necessary: they have cogently demonstrated that many of the widespread presuppositions about what young children know and what they do not know are erroneous, and that better insight into how children come to "do mathematics" should greatly improve the the teaching of arithmetic.