Take a second/minute and read this article by Andrew Gelman. All teachers, and especially all teachers who are scientists (or, at least, science-y), should think about how to evaluate and improve their methodology.
You could randomly assign students to different experimental groups and compare outcomes. This is logistically difficult for many of the most important hypotheses you’d (well, I’d) like to test, but it is feasible for some interesting questions. For example, how should we weigh various factors in determining final grades? You could make homework worth 10% of the final grade for half the class and 30% for the other half. Do the groups perform similarly well on exams? On the final? Are their final grades similar? Or maybe make attendance 0% for one group and 10% for the other (I’d be uncomfortable making it higher than that!). Do attendance patterns change? Do grades change? Or assign different homework (or group work), one emphasizing rote calculation, the other critical thinking and applications (I really like this one!).
We should also pre-test for general math skills (of both knowledge assumed in the class and latent talent). While we’re at it, post-test for them, too! Did you manage to teach your pre-calc students how to add fractions? Find out! And then tell the rest of us how you did it!
It would be difficult to test teaching styles or explanations of topics, though things could be done differently for different classes (keeping in mind that different classes aren’t necessarily exchangeable!). Tying this in with Kaisa’s previous post (well, not really), MOOCS (or remote-learning more generally) would allow this sort of experimentation: you could actually send each half of the class different videos!
There’s a lot more that can be said about this (especially how we can engage with the already-extant literature on effective teaching), but I’ll leave this as a start.