Requirements for public preschool teachers are generally more stringent than those for private preschool teachers. Some States require a bachelor's degree in early childhood education, while others require an associate's degree, and still others require certification by a nationally recognized authority.reschool teachers usually work their way up from assistant teacher, to teacher, to lead teacher-who may be responsible for the instruction of several classes-and, finally, to director of the center. Preschool teachers with a bachelor's degree frequently are qualified to teach kindergarten through grade 3 as well.