Curtis was born in Ada, Oklahoma. He attended Paschal High School in Fort Worth, Texas and entered UT in the fall of 1920. Curtis was a letterman at defensive end for the Longhorn football team for two seasons from 1922–23 and at guard for the basketball team for three seasons from 1922–24. In his senior year, Curtis played on first-year football and basketball head coach E. J. "Doc" Stewart's undefeated football (8–0–1) and undefeated Southwest Conference champion basketball (23–0) teams. The 1924 Longhorn basketball team received a retroactive national ranking of No. 3 in the Premo-Porretta Power Poll. Curtis received All-Southwest Conference honors in basketball following his senior season; he was also recognized as a consensus first-team All-American in basketball—one of only seven Longhorn men's basketball players ever to receive that honor, as of 2015.
- "Men's Hall of Honor – Albert S. "Abb" Curtis". texassports.com. http://www.texassports.com/aa.aspx?hid=75. Retrieved May 5, 2015.
- "Texas High School Basketball Hall of Fame" (PDF). drbillywilbanks.com. p. 54. http://www.drbillywilbanks.com/basketball/TX-HS-BASKETBALL-HALL-OF-FAME.pdf. Retrieved May 7, 2015.
- "2014-15 Texas Basketball Fact Book" (PDF). TexasSports.com. pp. 144. http://texassports_com.s3.amazonaws.com/documents/2014/11/12/2014_15_Texas_Basketball_Fact_Book.pdf. Retrieved May 5, 2015.
- 2014-15 Texas Basketball Fact Book, pp. 65–66
- ESPN, ed. (2009). ESPN College Basketball Encyclopedia: The Complete History of the Men's Game. New York, NY: ESPN Books. p. 536. ISBN 978-0-345-51392-2.
- 2014-15 Texas Basketball Fact Book, pp. 140–41
- 2014-15 Texas Basketball Fact Book, pp. 143