He played in Major League Baseball as a shortstop from 1978 to 1982 for the Texas Rangers and the Pittsburgh Pirates.

He made his way through their farm system quickly, and by the end of the 1977 season he was playing for the Triple-A Columbus Clippers.

Norman was traded along with Al Oliver from the Pirates to the Rangers in the first four-team blockbuster deal in Major League Baseball history on December 8, 1977 that also involved the Atlanta Braves, New York Mets and a total of eleven players changing teams.

The Mets got from the Braves Willie Montañez and from the Rangers Tom Grieve and Ken Henderson who was sent to New York to complete the transaction three months later on March 15, 1978.

In mid-May, starting shortstop Bert Campaneris was injured, and Norman was called up to the majors to back up replacement Jim Mason.

He played three innings in the field, handling one chance on defense, before he was lifted for pinch hitter Mike Jorgensen.

However, by the end of April, he had lost the job to Pepe Frías, who had been acquired from the Atlanta Braves over the winter.

In May, the Rangers signed Bud Harrelson to back up Frías, and Norman, who was hitting .219 in 17 games, was sent down to the minors.

After hitting just .186, he was traded during the offseason to the Montreal Expos for fellow minor league veteran Steve Baker.

After his playing career ended, Norman remained in the Expos organization as a minor league coach.

From 2008 to 2010, he was the manager of the Dominican Summer League Cardinals, although he was replaced for part of 2009 by Claudio Almonte.