In a new NBER working paper, evidence from random assignment of students to professors:

Introductory course professors significantly affect student achievement in contemporaneous and follow-on related courses, but the effects are quite heterogeneous across subjects.

While wondering whether to take that course with the visiting Harvard professor…

We find that the academic rank, teaching experience, and terminal degree status of mathematics and science professors are negatively correlated with contemporaneous student achievement, but positively related to follow-on course achievement.

And teachers, in case you ever wondered whether you should go easy on your students to boost your evaluation scores…

Across all subjects, student evaluations of professors are positive predictors of contemporaneous course achievement, but are poor predictors of follow-on course achievement.