I think there are a lot of reasons that being a researcher makes me a better teacher, but it's not a one-way street: teaching is also good for my research.
Teaching reminds me that I won't be stuck forever. I'll be honest, I remember being really confused by change of basis matrices when I first met them. I also recall fighting hard with the Isomorphism Theorems for groups, but happily, both these topics seem completely transparent now. I can simultaneously be sympathetic to my students' struggles and assure them that they'll get there eventually. This is an honest assessment born of experience, not feel-good propaganda, and yet it's much easier for me to remember this in the context of teaching. In research, you spend all your time dealing with the parts of the problem you don't yet understand, so it's really encouraging to have reminders that progress is possible.
Having to get to lectures on time also reminds me how nice it is to work on maths in a cafe!
