I've been reading Knowing and Teaching Elementary Mathematics for several weeks now, and finally finished it. This was a worthwhile read. The book is Liping Ma's doctoral dissertation. For her research, Liping Ma compared the "Profound Understanding of Fundamental Mathematics" between American elementary teachers and Chinese elementary math teachers. The results are fascinating and disturbing and certainly interest me in how math is taught in our homeschool.
The American teachers were mostly proficient in the algorithms to solve the problems set before them, the Chinese teachers were able to explain why the algorithms work (mathematically) and several ways to solve problems that didn't rely upon the standard algorithm. It makes me want to go and study arithmetic carefully, with special emphasis on the proper mathematical terminology.
This was not an easy read, between the technical language, the scientific nature of the presentation, and the disturbing lack of knowledge many of the American teachers showed. Some of the math I found difficult (particularly the chapter on dividing by fractions).
My main quibbles with the book are the sample sizes (the number of Chinese teachers interviewed is much larger than the American, but, really, that just gave them *more* chances to get it wrong). Also, she never seems to address the larger point that I kept seeing: the American teachers taught each subject, were generalists; while the Chinese teachers were Math teachers, specialists. This disparity in specialization would account for many of the issues detailed in the book: teacher study time (4-5 hours of preparation for 45 minute lessons. Teaching the same lesson 4-5 times a day to several classes of children. Office hours. Prep time with teachers in the field. Etc.)
I borrowed this from Inter-Library Loan, but it deserves more attention than I've been able to give before it is due back, so I hope to one day have it gracing my shelves. I would highly recommend this to anyone teaching elementary mathematics. Then you, too, can wish there was a book that laid out Mathematical Knowledge Packages in a systematic order for homeschool moms.