You need to .

Don't forget that to get the trophy and make the nursery operational you must

I don't think so. The Official Guide (which is usually reliable) doesn't say so. But it's probably irrelevant, since the Nursery School is less urgent than which DOES require a certain level of Construction.

Also (and I wish someone would put the link in, because my Search hasn't found it again), people have wondered aloud what skills are gained and came to no conclusion. MY conclusion is that the kids probably gain on the skill where I've checked a box for them. Since in the first place I checked those boxes to match the minimal skill they'd inherited, it's possible that inheritance controls. I've just gone thru and given everybody a different check, so I'll let you know what I find out.

Unless somebody beats me to it, of course.

And one more amusing thing about nursery school teachers: Unlike other adults, they frequently run--admittedly for short distances--even if "running" isn't in their Likes. This is probably an absolute necessity for administering the school.

That conclusion was false, and in fact inheritance controls.

Proof: We're educating triplet boys, and when they started school I checked Building (their highest skill) for one of them, Farming for another, and Research for a third. They have all gained identically, and the overwhelming bulk of those gains is in Building.

A perfect example of correlation not being causation: In other cases, I marked their highest skill, and they gained in their highest skill.

I don't know if anybody here writing the same - my search was without success.
Try this:
If you mastered your villagers can fish, build the hospital (healers are trained) and the nursery school - then put some villagers with various master skills or a scholar to the nursery school ... they will teach the children a short time but come back later ... If you don't play the game set the game speed on "SLOW". Your children will gain every skill when you return after some (8/10/15/) hours and are apprentice in one skill - it's amazing!

Does the thing they study have an impact on the skill they learn?

It doesn't seem to. I think it's randomly chosen.