But in the years since, he has become an even bigger superstar. Last year Wallace returned to see what Perlman was up to.
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At age 35, Perlman was performing more than 100 times a year, all over the world.
He began his career at the age of 3, when he asked his parents for a violin. The next year he came down with polio. Ever since he has had to wear braces on his legs and walk with crutches.
In 1958, when he was 13, he performed on The Ed Sullivan Show.
As a child prodigy, Perlman said, he was very aware of his disability. "I could show you reviews when I first came to the United States," he remembered. "'Handicapped violinist pretty good, despite disability.'"
"Or 'Crippled blah, blah, blah, dah, dah, dah. And as he went on the stage, hobbling on his shining aluminum crutches and very heavily sat down, but afterward we forgot all about it and it was just music.' And so on. And, every, every single review had to mention that," he added.
Perlman didn't like this kind of attention and thought it took away from the music itself.
He plays a wide range of music, recording country, ragtime and jazz in addition to classical.
Even so the violist wants to conduct someday, he said. "You have the sense of power" about controlling the "mass of sound" that an orchestra produces, he said. But before he conducts, he must accomplish more playing the violin, he added.
Perlman has also become known for his sly sense of humor. Over the years, Perlman has said that the violin is a Jewish instrument. When Wallace asked him why so many top violinists - Jascha Heifetz, Yehudi Menuhin, Isaac Stern, to name a few - were Jewish, Perlman was ready with an answer:
"You see, our fingers are circumcised, which gives it a very, very good dexterity, you know, particularly in the pinky," he said, breaking up with laughter. "I don't know what it is. Maybe it's a tradition."
Perlman is also famed for his ability to perform multiple tasks while practicing. Andre Previn said he came in one day to watch Perlman, and found the violinist practicing and simultaneously watching I Love Lucy on television.
"Actually, the best show to practice on is baseball," Perlman said. "It's terrific, because you can turn off the sound, and you know what's going on, and you practice your technique."
"I'm not talking about practicing thinking or anything like this," he said. "That's a totally different thing. I did some of my greatest practicing when I was in London watching cricket. It's a very, very low game. That's when you practice."
In the Perlmans' spacious New York City apartment, everyone practices. He and his wife, Toby, met 16 years ago when they were both studying the violin. They've been married for 13 years and have four children.
The Perlmans do their best to create a typical environment for their children. "Children aren't allowed to watch television," said Toby Perlman, with a twinkle in her eye.
"They have to make their own beds; they have to do the dishes; they have to do their homework. Mommy's at home when they get home from school, supervising with a whip in one hand and a gun in the other. Dinner is served on time," she said.
But imposing rules on her husband was much more difficult. The price of fame, according to his wife, is an increasing lack of privacy.
Nevertheless, said Toby Perlman, her husband is not a typical superstar egomaniac. "I think he thinks a lot about himself, but not a lot of himself."
Her husband agreed. "Sometimes there is a danger that with all the lights and all the success and the cameras and all the articles and the things, (you) forget what you - why you are here to begin with," he said. "And that is because you are playing the violin."
Twenty years later, Perlman still plays 90 concerts a year around the world. But as he hinted that he would in 1980, he has also started a new career: conducting.
In 1997 he began conducting orchestras, in places ranging from Tel Aviv to Philadelphia. Recently, he conducted his daughter Navah, 30, who is now an accomplished pianist.
And being the maestro is not quite what he thought it would be, Perlman says. "It's not easy," he says with a laugh. "And it doesn't give you a sense of power. I think that the sense of power...comes if what you want to do comes out. And it doesn't always come out unless you work."
Conducting, Perlman says, is much more like teaching than he had realized. "They would like to follow," he says. "And you want to make sure that what comes from your hand is something that they can follow."
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Five years ago, he and his wife started a music camp called the Perlman Music Program. Each summer 35 promising young musicians from around the world come to their camp to play music and to learn from the master.
His wife says that he is a gifted teacher. "The kids know that he's a superstar," she says. "And they're terrified. They sit down at their places n orchestra, and he begins the rehearsal, and he's a funny guy. And he's a punster. And he's sweet, and very concerned with how they feel."
"And within a few minutes, you can see the shoulders go down. You can see the expression open up. And they're off and running," she adds.
Toby Perlman launched the camp not only to erase her own childhood memories of pressure-filled music schools, but also, she says, to help her husband regain a sense of the joy in music.
"I took a look five summers ago," she says, "and saw this guy who never stopped complaining, had stopped smiling, was eating himself into oblivion."
"And suddenly during the first (camp),...there was the guy that I married. He was smiling all the time, and he was busy and he was running, and he was working, and he never complains," she says. "He's never tired. There's no such thing as a problem. And he's smiling all the time."
Broadcast story produced by Barbara Dury; Web story produced by David Kohn;