There's a British usage of "keeping term" which implies that the school is in session and the pupils ought to be there. Here's a link to a University usage:-

http://www.dur.ac.uk/resources/university.calendar/volumei/current/regs.keeping_terms.pdf

Come to think of it, there's a Scots usage "to keep school", "to keep church" that implies that he who does so is a regular attender. Perhaps that also gets turned round so that the school "keeps" implies that the school is open so the the laddy who "keeps school" can attend.

I've read it often enough, but not heard it or used it. I take "I don't care if school keeps or not" to be simply an emphatic form of "I don't care", the distinction between "school's in" and "school's out" being a fundamental one, at least to children.

I remember hearing the phrase "so and so kept school" on old TV shows. The settings were usually small towns or rural areas in the mid-West and West, if I remember correctly.

I always assumed from the context that the phrase meant something like "taught class in a one-roomed school house". A google search turns up at least two examples that seem to confirm this interpretation, albeit in New England settings; one from a diary kept by a woman from Waterbury Conn. in 1819, and another from a history of Northfield Mass. in the 1770's. I'm sure a more thorough search would turn up more recent examples. (Sorry not to provide the links. I'm not sure how to do it on this site.)

I grew up in urban California. I've never used the phrase myself or heard it used except on TV.

Well, "kept school" is a different construction, so I don't think it counts here. Interesting that John Cowan has "read it often enough"; I wonder how I've managed to miss it? It's a striking enough phrase I'm pretty sure I wouldn't have ignored it.

DARE has an entry for this construction, "in phr _not to care whether school keeps or not_ and var; Fig: not to care what happens; to go one's own way. _somewhat old-fash_". They have examples from 1852 onwards. DARE doesn't give this any regional labelling; the three post-1930s examples are from Pennsylvania, Maine, and Michigan.

Thanks, I was wondering if it would be in DARE.

Never heard it, never heard of it. Northern California, mother's family came in the Gold Rush - immigrated from upstate New York, previously from Massachussetts (1800 or so). The other side of that side came from France about the same time. The other other side is Irish, midwestern. Never heard the expression.

Via Google books, an earlier citation than your 1845 one is this 1841 arithmetic book: "If the school keeps six hours every day, how many hours will a pupil be in school during six days?"

And this one from 1832.

Here's a timeline of news items using "school keeps or not", showing that it starts up mid 1800s and peters out after the 1960s. (Click on blue bars for specific citations.)

Wow, that timeline is great—what a resource!

how to keep a castle or be it stand in the castle keep

"Experience keeps a dear school, but fools will learn in no other."

Attributed to Benjamin Franklin.