Answer by taking a quote from the following article:

Kenneth Duva Burke (May 5, 1897 - November 19, 1993) was an American literary theorist, as well as poet, essayist, and novelist, who wrote on 20th-century philosophy, aesthetics, criticism, and rhetorical theory. As a literary theorist, Burke was best known for his analyses based on the nature of knowledge. Furthermore, he was one of the first individuals to stray away from more traditional rhetoric and view literature as "symbolic action." Burke was unorthodox, concerning himself not only with literary texts, but with the elements of the text that interacted with the audience: social, historical, political background, author biography, etc.

He was born on May 5 in Pittsburgh, Pennsylvania, and graduated from Peabody High School, where his friend Malcolm Cowley was also a student. While he attended Ohio State University to pursue courses in French, German, Greek, and Latin, he later dropped out to move closer to New York City where he enrolled at Columbia University. During his time there, he was a member of the Boar's Head Society. Due to the constraining learning environment, however, Burke also left Columbia, never receiving a college diploma. In Greenwich Village he kept company with avant-garde writers such as Hart Crane, Malcolm Cowley, Gorham Munson, and later Allen Tate. Raised by a Christian Science mother, Burke later became an avowed agnostic.  In 1919, he married Lily Mary Batterham, with whom he had three daughters: the late feminist, Marxist anthropologist Eleanor Leacock (1922-1987); musician (Jeanne) Elspeth Chapin Hart (b. 1920); and writer and poet France Burke (b. ~1925). He would later marry her sister Elizabeth Batterham in 1933 and have two sons, Michael and Anthony. Burke served as the editor of the modernist literary magazine The Dial in 1923, and as its music critic from 1927-1929. Kenneth himself was an avid player of the piano. He received the Dial Award in 1928 for distinguished service to American literature. He was the music critic of The Nation from 1934-1936, and was awarded a Guggenheim Fellowship in 1935.  His work on criticism was a driving force for placing him back into the university spotlight. As a result, he was able to teach and lecture at various colleges, including Bennington College, while continuing his literary work. Many of Kenneth Burke's personal papers and correspondence are housed at Pennsylvania State University's Special Collections Library. However, despite his stint lecturing at Universities, Burke was an autodidact and a self-taught scholar.  In later life, his New Jersey farm was a popular summer retreat for his extended family, as reported by his grandson Harry Chapin, a contemporary popular song artist. He died of heart failure at his home in Andover, New Jersey.

Where did he teach the most?
teach and lecture at various colleges, including Bennington College, while continuing his literary work.