Young Children Reinvent Arithmetic: Implications of Piaget's Theory by Constance Kazuko Kamii with Georgia DeClark (Teachers College Press, 1985) discusses the teaching of mathematics to young children. This book review discusses the first edition from 1985, the one written with DeClark; a subsequent edition, written with Leslie Baker Housman, was published in 1999; Kamii has also written books detailing her research regarding children in second and third grade.
While it would be helpful to have a bit of knowledge about Jean Piaget's theories before reading the book, the book does include a few chapters detailing Piaget's theories of learning in young children, social interaction, and autonomy. The book also spends a few chapters discussing learning objectives for students in first grade (number concepts, addition, subtraction), which serves as a good introduction to what follows.
Reading through the first three chapters, it seems reasonable to conclude that a Piagetian would find the standard way of teaching young children arithmetic, through worksheets and drills and flash cards, to be inadequate. Kamii suggests that the best way to teach children these objectives would be through situations in daily living that would allow children to structure and define math problems from real-world ambiguities. Kamii suggests several such situations, but these sort of situations don't happen frequently enough in classrooms. So, Kamii's idea is for children to learn these objectives through group games that interest them. These games aren't highly involved; many of the ones described in the book are relatively simple or are variants on commercially-available games that provide for opportunities to use basic arithmetic. For a Piagetian, group games provide an avenue for structured play, social interaction, and development of autonomy, allowing children to learn as they play.
The book also discusses the experiences of DeClark's students over the two years that these ideas were tried in her class. This appeared to be a learning opportunity for DeClark as well as the students. The children appeared to enjoy this method of learning arithmetic. As for the results of this teaching approach, the results as given in the book are somewhat mixed (I haven't yet read any of Kamii's subsequent papers or books, so I don't know whether they present more concrete data). With such a small sample size (one classroom) and lack of controls, Kamii rightly finds it difficult to draw conclusions. The evidence presented does seem to suggest that the first-graders had at least as good an understanding of arithmetic as their counterparts who learned through drills and worksheets, and various anecdotes presented in the book also suggest that the children who learned mathematics through games were ahead when it came to developing number sense; for example, it is mentioned that they are able to find novel ways to solve problems they don't know and, while they make mistakes, almost never give nonsensical answers (an example of such a nonsensical answer would be "4" to the question "4 + 3"; when you add something to 4, the answer has to be larger).
Young Children Reinvent Arithmetic is a well-written book and is an interesting read. Not being an educator myself, I've never tried to implement these techniques myself and so can't speak for their efficacy, although there appears to be several benefits to this approach. Recommended reading for teachers of young children and homeschool parents of young children.