The correct answer to this open question is the following.
The kind of learning you think king Archidamus would consider “useless matters” is any kind of intellectual learning.
This is so because king Archidamus and the Spartans were true warriors. They trained men since they were children under severe conditions to became fierce warriors and defend Sparta from intruders. The war techniques were inherited from their ancestors and the king considered it important to maintain them. He also thought that being a warrior was an honorable thing