Schools quite often can change a visitors' visa into a Z class visa in China. Whilst this is slowly getting harder to do, with some provinces stopping this practice, it is still fairly common.
The important thing to remember is that they shouldn't send your passport out of the province. You can only get a visa for the province you are in, in the province you are in.
However, check with the other teachers there if you can and make sure they have a Z-Class visa. Some schools say they can, but really they can't.
The Z class visa shouldn't take long to do, though they will need your passport they shouldn't have it for more than a week, two at most.
YOU should keep your Z-class visa when they get it. Don't let the school con you into thinking they should keep it. You also need it to leave the country, so don't go without it, or you will be in big trouble when you try to leave!
Get these things in writing before you go!!!!!!!

Moderators/board admin: How about a *sticky* post for visa questions? I would be happy to write a guide for China.