The Master Patterner lives in the Immanent Grove and teaches senior students.
What exactly does he/she teach?
I don't know of any detailed discussion of the Patterner's teaching in published works. Whether Le Guin ever commented on it in interviews or articles I don't know.
The only story that covers the Patterner in any detail is Dragonfly in the anthology Tales From Earthsea. Irian stays in the grove for a half-month or more of the hot days of summer and the story includes some of her conversations with the Patterner. The impression I get is that the skill of the Patterner is to hear the trees. For example the Patterner says to her:
When she asked him if students came there from the Great House, he said, “Sometimes.” Another time he said, “My words are nothing. Hear the leaves.”
My guess is that to hear what the trees are saying is to understand Segoy's plan for the world. When students come to the grove the Patterner tries to teach them to listen to the trees so they will understand the world and their part in it.
The Immanent Grove is one of the Old Powers, and is said to mingle its roots with all forests and woods in Earthsea. It is a mystery of connection, of integrity. As a symbol, it resonates with other Le Guin works, particularly The Word for the World is Forest.
The novella, "The Finder" in Tales of Earthsea introduces us to the first Patterner, a woman named Ember, who declines to explain to the protagonist, Otter, what the Grove is, saying that can only be learned from the Grove itself.
Patterning seems to relate to the balance, to Equilibrium, how every life and every act of magic affects every other. Since the magic of Earthsea can only be understood in the context of that world, it is hard to relate to the philosophy or science of our own world, but one avenue of further reflection may lie in knowing Le Guin also prepared her own translation of the Tao Te Ching.
I think it's worth noting that when Ged joins the two halves of the ring of Erreth-Akbe, he is said to do so 'not by using binding spell like with an old kettle, but using patterning'(cited from memory); meaning, patterning is not only 'listening to the wisdom of the trees'; but also a way to manipulate things; probably at the innermost level of their being.