An examination of the day-to-day work of high school teachers shows that fundamental ideas and results from abstract algebra, linear algebra, and number theory are useful tools for teaching high school algebra and, more generally, for bringing coherence and texture to much of the high school curriculum. Furthermore, these ideas and results are useful in lesson planning, task design, and in other “out of classroom” activities essential to the profession. The talk concludes with some suggestions for how undergraduate mathematics courses can make explicit connections to these aspects of mathematics teaching.