His four sons Butch, Craig, Bill, and Dick all became golf professionals and had a keen eye for teaching, just like their father.
And from a young age they knew that Claude's golfing prowess opened some pretty special doors.
"My father played with so many presidents himself," Harmon explained. "He played with Eisenhower and he played with Ford, Nixon, and I got to play with a lot of them too when I was younger.
"I think if my father was still alive he'd be very proud of all of us. He'd be proud of the four sons, he'd be proud of his grandson because in reality we're just carrying on what he taught us and what he did and we're a very proud family.
"He had the greatest eye I think of anyone that's ever taught and he saw things that other people didn't see and quickly, he knew how to fix them, he'd watch you hit five or six balls and he'd know exactly what you were doing wrong."
It was that wisdom passed down from father to son that led Harmon to believe that the 13-year-old kid presented to him in 1993 could go on to be one of the greatest players the game had ever seen.
Woods' father Earl had noted the work Harmon had done with Greg Norman and asked if he could bring his son over to hit some balls. The session was filmed by a Butch's young son Claude.
"You've seen the footage, he's a tall skinny kid in tennis shoes hitting balls and you just saw this natural talent that this kid had," Butch said.
"I asked him so many different kinds of questions about how do you do this? Why do you do that? He didn't know it at the time but I was giving him an examination, trying to find out really what ticked in his head, what he knew about golf.
"The kid was special, he was just a special person you could see it. It was no surprise to me that he turned out to be who he was."