It is normal, the school should help you to apply for the work visa, he should apply for work permit and visa notification letter for you. I guess that the school is not qualified to employ foreigners.

For China work visa application, Foreign experts, scholars, teachers or managers shall provide the original version and photocopy of the Employment Permit from the Ministry of Labour or the State Administration of Foreign Experts Affairs of China together with formal invitation letter from Government-authorized company, university or relevant local authority.

Thanks for replying Nancy. Do you know if I need to show proof of a round-trip ticket? I'm concerned because I only have a 1-way

you don't need to provide the proof of round-trip ticket in most situations, in certain situations, the visa officer will ask you to provide it. Please don't worry.