As I'm starting to apply to positions which require a "Teaching statement" or "Pedagogical thinking" section, I thought I should go see what evidence-based research has to say about teaching university level mathematics.

For now I found this recent literature review, dealing with this precise subject:
https://www.sciencedirect.com/science/article/pii/S0732312322000542

I hope it gives me some insights.
I have never been explicitly taught how to teach, as were most of my colleagues and in fact also my teachers. We learned intuitively via modelling - we teach the way we were taught ourselves. Is that good? Maybe, maybe not, that's a good question. Modelling is how we learn many things.
One might argue: if this method brought us to where we are (i.e. people who know math very well and can do mathematical research), then it's probably good.
A counter-argument would be - maybe it was good for us because we were specifically suited to this method, but we are missing others which do have the potential, but this method is not right for them.

Well, I don't know the answer, I hope there are some evidence-based conclusions I can draw from the above paper, and would love to hear your thoughts and experiences on the matter.

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Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn

There has been significant growth in the literature base exploring questions of teaching in undergraduate mathematics. In this paper we synthesize the…

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