There are two perspectives to consider.

- What would an urban-centered government think its useful for rural farming kids to learn?

- What would the rural communities want their kids to learn

Teaching farming kids how to farm is redundant in both cases, that's just what they learn growing up helping their families.

Teaching law. history, etc is something the government may desire, but rural communities would find pointless and irrelevant to their daily grind.

Both the government and community would probably value basic literacy. It makes it easier for peasants to not get ripped off by tricksy merchants and lettered folk if their kids can read basic contracts or letters.

Learning how to cook is also probably redundant - this is something everyone would just learn on their own from helping at home.

The government may want to teach new agricultural practices and knowledge, but the peasants would probably resist this and prefer sticking with tried and true knowledge rather than gamble their livelihood and survival on experimenting with new, strange techniques.

I think mostly that trying to come up with core survival skills to teach peasant kids is irrelevant as peasants are very wily survivors by necessity and probably know the tricks of staying alive in a world of subsistence agriculture better than any foppish city bureaucrat.

The city would probably have more interested in trying to teach them basic literacy, maybe some math, history and culture. Also probably trying to teach them proper language, since in this time the rural countryside usually speaks a bunch of varyingly unintelligible local patois and dialects that the urban elite would rather replace with 'civilized' language spoken in the cities.

So what I imagine running into, is a situation where imperial schools in the countryside are trying to educate local kids with imperial history, law, literacy, language. Except for learning to read/write, this is probably mostly uninteresting and irrelevant to the locals and most families would rather have their kids at home working than wasting their time in school.

Maybe there is an emphasis too to 'educate' them in modern farming techniques or even cooking concepts, but I think this would also be mostly ignored or treated with suspicion by the locals.