There were so many lessons to learn in Avatar: The Last Airbender that quite often, the teachers also became the students. Throughout their journey, the Gaang trained with the great masters of the Order of the White Lotus, and along the way, they changed the lives and perspectives of everyone.
Everyone, that is, except for the cackling mad King Bumi. Much like the element he wielded, Bumi remained resolute and unchanged from start to finish, and this may offer some insight into the Earthbender's wild talent.
Aang and his newfound friends from the Water Tribe, Sokka and Katara, crossed paths with King Bumi early on in the first season with the fifth episode "The King of Omashu." When the trio first ventured to the Earth Kingdom city, which is known for its creative courier system, Aang reminisced about a friend he knew there over 100 hundred years ago that was a mad genius. When the Gaang held an audience with the King of Omashu, he subjected the Avatar to tests of creative thinking, only to reveal he was Aang's childhood friend, the powerful Earthbender known as Bumi.
What did not stand out as odd at the time but is starkly peculiar in retrospect is that Aang was the only character to learn a lesson from the encounter. Bumi administered his tests and waited with a quiet confidence that the others would catch up, and Aang's lesson in learning to look at the world differently would be one he carried through the rest of the series. He would cross paths with Bumi on various other occasions, trying to free him when Omashu was later captured and learning he freed himself when the White Lotus arrived to liberate Ba Sing Se, but at no point did Bumi go through an emotional arc or learn a lesson.
But this isn't the case for his fellow members in the Order. The first they'd meet is Iroh, Zuko's uncle, who fully embodies the archetype of the wise sage but who nevertheless went through an emotional journey all his own. Over the course of his relationship with Zuko, he sacrificed his heritage, turned from his nation and discovered his true destiny in Ba Sing Se. Even his brief encounter first with Toph left Iroh a valuable lesson, as Toph made Iroh realize that Zuko needed to know how much his uncle needed him. He never stopped learning.
The three other members of the White Lotus learned their lessons in the stories that introduced them. Jeong Jeong, Pakku and Piandao all refused to take on new students when Aang, Katara and Sokka presented themselves for training. But each was reminded of the value of their knowledge and the responsibility they had to the world to pass it on. Pakku was even so affected he broke from the customs of the Northern Water Tribe and journeyed across the world to rediscover the love of his life. But Bumi? Bumi never learned a thing.
Whether they were lessons in creative thinking or learning when to wait and listen, Bumi always had advice of his own to dispense and little to learn. Unlike fire and water, which are ever-shifting and in a constant state of flux, Bumi embodied his element by resolutely resisting change. It's notable that he's the only White Lotus member the audience sees as a child, and even then he had the same cackle and snort and lessons to give.
It's valuable that Avatar went out of its way to show that even the wisest among the older generation still had something to learn from the younger heroes, but it's a genius stroke of character work to have Bumi embody the contrary point to that lesson. Not everybody needs to change and some people never do. That doesn't mean such people are always stubborn or obtuse, either. Sometimes they're just a mad genius.