Norman played basketball at UCLA and was co-captain during his senior year, when he was named first-team all-conference in the Pacific Coast Conference (PCC).

[2][5] In Norman's sophomore year in 1949–50, UCLA qualified for the NCAA tournament, the first in Coach John Wooden's career.

[8] In 1951–52, Norman was co-captain of the team with fellow senior Don Johnson,[9][10] and the Bruins won the PCC title and qualified for the 1952 NCAA tournament.

[7][14] Wooden asked him to moonlight as the coach of the freshman basketball team, and Norman compiled a 94–22 record in six seasons while successfully employing a zone press.

[17] Wooden began coaching UCLA in 1948–49, but the Bruins never advanced past the first round of the NCAA tournament until after Norman was hired.

[1][2][19] The Bruins had employed a man-to-man press with some success in 1962–63, but it had been ineffective in their opening-game of the NCAA tournament—a 93–79 blowout loss to Arizona State.

[20][21] Norman instead proposed a 2–2–1 full-court zone press,[22] which quickened the pace of the game and was influential in the first two national titles won by the Bruins, who were undersized.

Under Norman, UCLA's recruits included Alcindor from New York, Walt Hazzard from Philadelphia, and Lucius Allen from Kansas.

[1] In a rout of Houston in the semifinals of the NCAA tournament,[1] Wooden credited Norman for devising the diamond-and-one defense that the Bruins used to contain Elvin Hayes, who was averaging 37.7 points per game but was held to only 10.

[16] Morgan promised Norman that if he stayed, he would become the Bruins' head coach once Wooden retired, which ended up being seven years later.