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We would need to compare private vs public schools but those could easily be more about the students than the teachers.
That said, the research is quite clear that unions hurt student performance, and if your only goal is improving student performance non-union charter schools seem to be the best option. By giving parents as much choice as possible, parents tend to pick better schools, and poorly run schools are force to lay off staff or go out of business, so better run schools can replace them.