Adding It Up: Helping Children Learn Mathematics
I wanted to pass on this book link to those of you who are interested in the pedagogy of mathematics: Adding It Up: Helping Children Learn Mathematics (2001) by Center for Education (CFE)
A big plus is that you can read the whole book online for free.
This book talks a lot about how the conceptual understanding of basic arithmetic develops in children during grades 1-8. It explains common misconceptions. The book cites a lot of research about how children learn addition, subtraction, multiplication, division, fractions, and proportions. It doesn't deal with geometry or measuring, because the authors decided to concentrate on "number" concepts.
All in all I think it's good basic reading for all of us who teach math. You can read it online by chapters. Here's a direct link to page 187 which starts dealing with single-digit addition. Here's a table comparing various addition and subtraction problem types. And here's a link to the discussion about single-digit subtraction.
Follow this link to the main page of the book, and down further you'll see the links to the various chapters in order to read it online. Or, see the widget on the right.