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For Normal school in Shanghai, I do not think they will hire a expat.

There is some international private school in Shanghai but normally requires very
high.

is the biggest renting website in china and good luck!

For public schools, there's little or no chance.
For private schools, that being said-expats driven schools, your friend has a chance to apply , hopefully get an interview and hired!

Shanghai? Why?

It's too expensive, too Western and far too difficult to get a position.

However, if you set your sights to a third tier city, you probably will be able to find a nice teaching position. Now, the rate will be lower, but so will be the cost of living. I know numerous expats that are doing quite well making low salaries in the more remote areas of China.

If you work for two years in a secondary of third tier city, you won't have any problem moving to Shanghai and finding work.

Best of Luck.

Why on earth would someone with solid math qualification and 10+ years experience  under his belly leave all the glory and chose a place where nobody would appreciate them.
Local education system is 100% Chinese. Not an option.
International schools require qualification and experience with US or UK curriculum. Possible but they have already a pipeline filled with candidates cut for the position.

Living cost: there's  no significant difference between 1-tier cities and n-tier cities. Maybe the difference is that those living a pipe-dream could keep living in denial while hiding in slum-like neighborhoods filled mainly with poor locals who would obviously become rich by 2025 according to local propaganda.