How do those engineers get their education? Do they find a mentor engineer? So for each engineering student you need an already engineer teacher?
Or would there perhaps be a school of engineering with a hierarchy to organize the engineering lectures so there could be more students per teacher?
But there's not only engineering. Perhaps we might also need medical schools, art schools, sewage maintaining schools. Maybe those schools might want to interact with eachother in order to provide consistent curriculums and aid students if they want to switch from one school to another. Perhaps we need a department of education to coordinate all this schools.
Maybe, like we arrived at the department of education, we might want departments for other matters. Look! A government!