SCOTTSDALE, Ariz. -- Travis Shaw has a bit of catching up to do. His beard isn't nearly as long as those "Duck Dynasty" jobs adorning the faces of his older brethren with the World Series champion Boston Red Sox.

"In their Minor League system, they want their players clean," Shaw said. "When the season ends, they can grow their facial hair."

Shaw hopes to scratch the dirt with his cleats and gain some firm footing at first base at Fenway Park some day, and he has been building his resume with the Surprise Saguaros of the Arizona Fall League.

Shaw has made significant strides with the Saguaros. In Week 5, he had a league-leading .538 batting average, .667 on-base percentage and .923 OPS with 12 total bases. Shaw had two doubles, one home run, two RBIs, five walks and five runs scored.

All of those added up to Shaw being selected as the league's Player of the Week. Glendale Desert Dogs left-hander Andrew Heaney was named Pitcher of the Week.

The 6-foot-4, 225-pound Shaw is a bit bigger than his father, Jeff, a relief pitcher who registered 203 saves from 1990-2001 with five teams, including the Cincinnati Reds and Los Angeles Dodgers.

Travis, 23, doesn't carry nearly the same expectations as other sons trying to become Major League players like their dads.

"I don't feel pressure," Shaw said. "He played a different position, and that's a big difference."

Shaw tried pitching briefly in high school, but showed more of a knack for knocking the baseball to all parts of the park as a third baseman. He was drafted by the Red Sox out of high school in Ohio (32nd round in 2008), but went to college to play at Kent State. Boston called again in 2011 (ninth round) and Shaw signed a contract. Almost immediately, the Red Sox moved him to first base.

"I wasn't surprised. I figured I might end up going that way even before they told me, so that was fine," Shaw said. "The first year [in the Minor Leagues] was a bit rough defensively, but I got used to the footwork and things worked out all right. People think you can just throw anybody over at first, but that isn't always the case."

It helped that the left-handed-swinging Shaw could hit. He did all right in the lower Minors, but struggled a bit, average-wise, when he reached Double-A Portland in 2012 and again in '13.

Shaw has a combined Minor League average of .256 with 43 home runs and 172 RBIs in 316 games. The extra time spent in the Fall League has helped him quite a bit, he said.

"It has been pretty good. My swing has been more consistent," Shaw said. "I had sort of become a little pull-happy, trying to hit home runs, but now my swing is a lot more natural and I am using the gaps more. I'm hitting more doubles, and if the home runs come, they come more naturally. I am starting to feel like I can handle any pitch in any situation."

Entering play on Tuesday, Shaw was hitting .352 with five doubles, five home runs and 17 RBIs. Overall, he was sixth in the league in hitting, second in home runs and tied for third in RBIs.

Shaw remembers hanging out in Major League clubhouses when his father played, and he occasionally will lean on his father's experience to help him along.

"It's nice that he has been there," Shaw said. "His advice has helped tremendously."

Shaw thinks he is likely to begin 2014 at Portland again.

"But I also think I can put myself on the map if I can move up to Triple-A," he said.

And beyond? David Ortiz and Mike Napoli aren't getting any younger and probably won't be with the big club a whole lot longer. Someone else eventually will have to step in.

"Hopefully, that can be me," Shaw said.