A native of Kentucky, Kirwan completed three degrees in mathematics, attending the University of Kentucky for his bachelor's degree and Rutgers University for his master's and doctorate degrees. Beginning in 1964, Kirwan was a mathematics professor at Maryland. After over 15 years on the faculty, including four years as head of the mathematics department, Kirwan joined Maryland's administration, beginning as chief academic officer in 1981. Kirwan had two stints as the interim president of the university in 1982 and 1988 before being formally elected by the Board of Regents as president in 1989, a position he would hold before leaving to become president of Ohio State University in 1998. Kirwan returned to Maryland in 2002 to serve as chancellor of the USM before retiring in 2015.