Reddit mentions of Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series)

I wonder how much of it has to do with how the arithmetic is taught though. I read Knowing and Teaching Elementary Mathematics even though I teach high school because I found it fascinating. Basically it contrasts US elementary teachers with Chinese elementary teachers and the difference in approach to teaching the basic techniques is pretty amazing...

Math content knowledge on a subject is pre-requisite to pedagogical knowledge. If a teacher can't divide fractions, xe won't be able to effectively teach xir students how to divide fractions either. This point was highlighted in Liping Ma's book that has been making the rounds in math ed community.

You are right that just knowing how to divide fractions doesn't mean that you can effectively teach it; that's where the jingle "ours isn't to ask why/ just invert and multiply" came from: well-meaning teachers who are trying the best they know how to get their students to pass the tests.