Teachers 

Since Peter, as only one, does not teach Dutch language, and Mr. Dutch does not teach any course that is taught by Karl or Mr. Painter, it follows that Peter and Mr. Dutch are the same person and that he is at least math teacher. Simon and Mr. English both teach history, and are among the three Dutch teachers. Peter Dutch therefore has to teach chemistry next to math. Because Steven is also chemistry teacher, he cannot be Mr. English or Mr. Painter, so he must be Mr. Writer. Since Karl and Mr. Painter are two different persons, just like Simon and Mr. English, the names of the other two teachers are Karl English and Simon Painter.

Summarized:
Peter Dutch, math and chemistry
Steven Writer, Dutch and chemistry
Simon Painter, Dutch and history
Karl English, Dutch and history.