This is a lie. Z Visas must be obtained first, and are immediately replaced by 1 year Resident Permits as well as the working expat being issued a Foreign Experts Certificate which is the legal requirement to work as a teacher (falling under the "Wen Jiao" category of "Experts")in China. An F Visa is the gray line where you put yourself at risk at the whims of local government officials who might do a "surprise" inspection at any time to check if you are working legally or not.

Also, don't forget that if you wanted to switch jobs in China without ever having to leave the country you will need a letter of release, and if you are on an F Visa that means you will not be getting one. The money you save now for the Visa will be money that is later spent on making that extra plane trip to Hong Kong or back to your country of origin. Think to yourself: Is it worth it?