The teachers in Sumerian schools weren't a formally defined profession like today. Instead, the role was often filled by
scribes who were already literate and skilled in the cuneiform script. These scribes acted as instructors, passing on their knowledge and skills to the next generation of scribes. They weren't necessarily specialized educators in the modern sense, but rather experienced practitioners teaching their craft.
The social standing of the teacher varied depending on their own status within the community. Some might have been associated with temples, others with palaces, and others still might have been independent tutors. But the key element was their mastery of cuneiform writing and the various associated skills (administration, accounting, literature, etc.) that they taught.