One of the most prominent and influential teachers here in the Northeast US was
John Swallow.
https://music.yale.edu/2012/10/24/in-me ... er-swallow
http://www.osborne-conant.org/John_Swallow.htm
He taught generations of students at New England Conservatory, Yale, the Hartt School, and others, and he was central to the growth of brass chamber music and contemporary music for trombone in the 20th century. I never studied with him myself but many of my colleagues did, and they tell me his teaching methods focused on the use of rhythmically and tonally challenging etudes such as Bitsch to develop flexibility and accuracy of rhythm and pitch. If I understand correctly, he would often discuss phrasing in terms of metric displacement and the mental flexibility to think outside the constraints of barlines. Many of his students I've played with are characterized (at least in part) by a very focused (but easy, not tight) core to the sound, outstanding control of soft dynamics, and very accurate pitch and rhythm.
One of Mr. Swallow's most prominent students is
Norman Bolter, founding member of the Empire Brass and retired 2nd trombonist of the Boston Symphony, who has built a tremendous legacy of students in orchestral and teaching positions around the world that I see as underrecognized and underappreciated (full disclosure: Norman was my teacher in grad school).
Just off the top of my head, a few of Norman's students include:
Doug Wright, Minnesota Orchestra principal
Jonathan Randazzo, St. Louis Symphony principal
Jim Nova, Pittsburgh Symphony 2nd/utility
Matt Guilford, National Symphony bass
Ross Holcombe, Florida Orchestra asst principal
Darren Acosta, Finnish Radio Symphony principal
Allen Meek, Singapore Symphony principal
Karna Millen, US Coast Guard Band principal
One of the things that makes his "school" of teaching difficult to appreciate is that his students rarely if ever sound like him or each other. Chatting with my classmates, I realized my lessons were completely different from them. He was always responding to the needs of the student in front of him; I had pretty clear musical conceptions and needed to sort out playing mechanics, so we worked a lot on Arban's and developing an efficient, focused aperture. Some of my classmates needed to do more work developing their musical imaginations and approach technical challenges from there. But he never tried to make anybody try to fit into any particular tone concept, working instead to develop and refine that particular player's tools.
And another master teacher that hasn't been mentioned is
Per Brevig, former Met Orchestra principal trombone who was the primary teacher at Juilliard before Joe Alessi and the main trombone teacher at the Aspen Music Festival for 50 years. He was also a major player in the commissioning of new music for trombone in the 20th century. Dr. (then Mr.) Brevig was one of my teachers early on, and he stressed absolute accuracy of pitch and rhythm and a very direct, straightforward method of tone production, blowing right into the center of every pitch.
One of his students was
John Kitzman, retired Dallas Symphony principal and an often unsung hero of taking excellent players over the finish line of refining their orchestral excerpts to win auditions.