In Taiwan, you can change your visa from a visitor visa to a resident visa which will then allow you to get your Alien Resident Card (ARC).
However, some visas may be marked that they are not transferable and thus can’t be changed. This depends on where and how your received your visa. In cases where you can’t change a visitor visa to a resident visa, one must receive a letter of employment from their school and a document from the Foreign Affairs office stating that your application to that school is valid. Your school should aid you in getting this document. Once these proof of intended employment documents are obtained, the teacher may leave the country and get a new visitor visa that can be transferable to a resident visa. The easiest place to get this done is at the Hong Kong Taiwan visa office.