Galatea Merrythought was teaching then.
As Dumbledore tells Harry, Galatea Merrythought had been teaching at Hogwarts for nearly fifty years at the time the Dark Lord asked Headmaster Armando Dippet for the Defense Against the Dark Arts when he was eighteen.
“Somehow, Harry knew the answer even before Dumbledore gave it.
‘Defence Against the Dark Arts. It was being taught at the time by an old Professor by the name of Galatea Merrythought, who had been at Hogwarts for nearly fifty years.”
- Harry Potter and the Half-Blood Prince, Chapter 20 (Lord Voldemort’s Request)
Since the Dark Lord would have been eighteen in around 1944, that’d mean Galatea Merrythought would have started teaching at Hogwarts in approximately 1894. She would have been teaching through the entire period between 1901 and 1908.