“What Khan represents is a model that’s tapped into the desire that everyone has to personalize the learning experience and get it cheap and quick,” said Jim Shelton, assistant deputy secretary for innovation and improvement at the Education Department.

Mr. Shelton predicted that there would be “a bunch of knockoffs” that would take the Khan approach and try to expand on it. “This is going to spread like wildfire,” he said.

Mr. Khan grew up in a suburb of New Orleans, where his mother, who is from Bangladesh, raised him on her own by cobbling together a series of jobs and businesses. He went to public schools, where, as he recalls, a few classmates were fresh out of jail and others were bound for top universities.

Math became his passion. He pored over textbooks and joined the math club. He came to see math as storytelling. “Math is a language for thinking,” he said, “as opposed to voodoo magical incantations where you have no idea where they’re coming from.”

The YouTube lectures got their start six years ago when Mr. Khan needed a way to help a cousin catch up on high school math. They are startlingly simple. Each one covers a single topic, like long division or the debt crisis, usually in a bite-size 10-minute segment. The viewer hears Mr. Khan talking, in his typically chatty, older brother sort of way. But his face is never seen, just his scribbles on the screen. More recently he has included two outside specialists to give lectures on art history topics like the Rosetta Stone and Caravaggio.

Today, the Khan Academy site offers 2,700 instructional videos and a constellation of practice exercises. Master one concept, move on to the next. Earn rewards for a streak of correct answers. For teachers, there is an analytics dashboard that shows both an aggregate picture of how the class is doing and a detailed map of each student’s math comprehension. In other words, a peephole.