The Residency Permit is the thing that is in the passport. There is no such thing as a "Working Permit". Many people (depending on their situation and job) may also have a "Foreign Expert Certificate" which is a separate book.
There is such a thing as an Alien Employment Permit. Teachers and other "experts" generally get the Foreign Expert Cert. Other life forms get the Alien Permit.
What do you have to do to get that then?
Eight years ago I had to look after foreign teachers holding different work permits:
a. Foreign Expert Certificate issued to teachers with a degree
b. Alien Employment Permit issued to teachers without a degree
The problem, as I see it, is that there have been significant changes in the regulations relating to residence in China since 2008. As a general rule a degree is now a prerequisite for those wanting to teach in China.
It may still be worth exploring the chances of getting an AEP as a backdoor option but I do not hold out much hope that it will be granted. I think they are given for tradesmen/technicians with needed skills.
A quick flick through Google may give an answer.