So, a few decades after a nuclear holocaust, the people of New Jerusalem out near the great salt lake open up a school to teach people. Since the war, however, technology is gone back to the dark ages, including the ability to mass produce books. My question is, can a school function without mass produced books?
The school has about 500 students
Mass production of books was made possible by Gutenberg's invention.
Before that books had to be hand written one by one, making them extremely valuable items.
Nevertheless there were schools before that time, mostly relying on oral transmission of knowledge and large libraries where the scholars could copy books needed for research.
The very same can happen in your case.
Adding to the answer by @L.Dutch: not only were there colleges and universities before the invention of the printing press by Gutenberg, but also was the transmission of knowledge from these times kep up for centuries to come. A lecture is called 'lecture' - derived from the Latin term for reading - because the lecturer would read the only available book or script of the knowledge to the students who were to take notes. For this process you do not require any printing press, but paper should be available. The main problem will be the change of knowledge transmission for students who grew up in 'pre-apocalypse' time when barely any knowledge has to be memorised any more, as it can always be looked up online on any smart phone anytime, anywhere, whereas afterwards knowledge has to be meomrised literally, as they can only 'look things up' in the library within their own mind.
Sure, they can stock places of learning with books the same way they did before there was a printing press: people copying books.
Eg. every student copies a book they have to learn twice, verbatim*: once for their own use and the second one goes into the library. And they are learning while they do it. The start would be slow but it would pick up exponentially afterwards.
*alterations of course are unavoidable but then, even modern text books have errors in them.
Depends on if paper is available.
For undergraduate level mathematics at least, it is fairly common a lecturer will produce their own online notes for a course rather than assigning a textbook. This is essentially writing a small book.
If paper is available, your lecturers should be required to produce such books. Then each student, as part of their degree work, is required to make manual copies of several books.
In fact you can pay scribes who are not students. They don't even have to understand what they're copying. Remember the apocalypse was only a few decades ago so literacy is much more common in your world than during the dark ages (for now).
Soon you have a respectable library.
If paper is unavailable then even students 'taking notes' will be impossible. This is an even bigger obstacle than the nonexistence of a printing press and warrants several questions of its own.
Bonus: I don't know how much easier it is to build a paper factory than a printing press. But the university should have its own paper factory and students are required to put in hours to manufacture all the paper they need, or pay extra fees for the labour.
In addition to all the general answers that have been given about the historical preservation and transmission of knowledge before the mass printing of books, you may want to copy a technique used when books were only somewhat rare: have a group of students share a book, sitting in a circle around it. The students always sit in the same relative positions so each one learns to read best at a somewhat different angle.
I can't find any references to this practice online but I remember reading about it in a book about Jewish schools in poor areas of Europe.
The same era (or a slightly earlier one) also spawned innovative fonts designed to reduce the amount of ink needed for any given letter.
The knowledge would be shared via oral stories. And, as consensus is developed for the oral tradition of a given story, it will be written down so the story stays consistent going forward.
You have a big problem, without cheaply and widely available books you don't have mass literacy any more, add to the lack of availability the fact that it's not worth learning to read when you're struggling to make a living farming without full mechanisation, and the fact that modern acid pulp paper breaks down within 5-10 years under ideal conditions. Taken together all that means that decades after the fall of civilisation you don't have teachers who work from exclusively books any more, maybe some notes but not huge libraries of textbooks. In short no you can't have schools for modern style book learning without books being readily available, pretty much from birth. You can still have training schools for practical skills though, it's amazing what that used to include before computers, including but not limited to architecture, surgery, and advanced algebra & calculus all taught verbally to be done without more than a temporary sketchpad.
