Armando Dippet is the Headmaster at this time. Albus Dumbledore’s career as a teacher lasts for several decades, starting in the 1920s. In 1927, he is reassigned from teaching Defense Against the Dark Arts to teaching Transfiguration after he angers the Head of Magical Law Enforcement. He eventually becomes the Headmaster in the mid-1960s.
Dumbledore taught Defense Against the Dark Arts for a number of years, but was reassigned as a punishment by Torquil Travers in 1927. From that point on, he apparently taught Transfiguration. He was the Transfiguration teacher when Tom Marvolo Riddle was a student (HBP13, HBP20). He taught for over thirty years before taking over as Headmaster in the mid-1960s.