For example, in the phrase "教师之为教." I know it is formal language, but what is a good way to understand it in English?
I searched for 教师之为教 and got a few hits suggesting it's the first clause from the sentence "教师之为教，不在全盘授予，而在相机引导。……".
In that case, 之 means of; from and 为 means act; perform.
教师之为教 literally means 'the act of teaching of a teacher (should be ...)', and in this context means 'the art of teaching (is ...)'.
The art of teaching is not to explain everything at once, but to guide at opportune times.
imo, "教師之為教" is an composed verse, pretending in old written style. so, here's the literary chinese approaches:
first, some examples of "之為":
once (一) is (為) too much (甚)
之 indicates the occurrence of "once", while 為 is "being"
understanding (知之) is (為) understanding (知之)
之 indicates the state of understanding
buddha (佛) 's (之為) doctrine (教), or
the doctrine (教) of (之為) buddha (佛)
teachers (師) ' (之為) principles (理), or
the principles (理) of (之為) teachers (師)
so, back to "教師之為教".
imo, it's ugly, "not-literary-chines-enough", a better saying is "師之為教".
i would suggest "之為" as 's:
teachers (師) ' 之為 teaching (教)
or "之為" as of:
the teachings (教) of (之為) teachers (師)
or, a more clumsy one:
the teachings (教) conducted (為) by (之) teachers (師)
since chinese language is fuzzy, there're a few more ways of explanations, the above are just some of them.
have fun :)