What Mathematical Knowledge Improves High School Math Teaching?

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Only recently has there been research on what sort of mathematics training is actually useful to teachers. I will begin with brief survey of this research, for both elementary and high school teaching. I will discuss some recent results on how policy tends to be more consistent with these findings at the elementary level than at the high school level, and why this may be. I will conclude with some open questions about the mathematical preparation of high school teachers, and the potential role of mathematicians in addressing these questions.