He can see that a girl sitting against the wall is zipping through geometry exercises; that a boy with long curls over his eyes is stuck on a lesson on long equations; and that another boy in the front row is getting a handle on probability.

Each student’s math journey shows up instantly on the laptop Mr. Roe carries as he wanders the room. He stops at each desk, cajoles, offers tips, reassures. For an hour, this crowded, dimly lighted classroom in the hardscrabble shadow of Silicon Valley hums with the sound of fingers clicking on keyboards, pencils scratching on paper and an occasional whoop when a student scores a streak of right answers.

The software program unleashed in this classroom is the brainchild of Salman Khan, an Ivy League-trained math whiz and the son of an immigrant single mother. Mr. Khan, 35, has become something of an online sensation with his Khan Academy math and science lessons on YouTube, which has attracted up to 3.5 million viewers a month.

Now he wants to weave those digital lessons into the fabric of the school curriculum — a more ambitious and as yet untested proposition.

This semester, at least 36 schools nationwide are trying out Mr. Khan’s experiment: splitting up the work of teaching between man and machine, and combining teacher-led lessons with computer-based lectures and exercises.

As schools try to sort out confusing claims about the benefits of using technology in the classroom, and companies ponder the profits from big education contracts, Khan Academy may seem like just another product vying for attention.

But what makes Mr. Khan’s venture stand out is that the lessons and software tools are entirely free — available to anyone with access to a reasonably fast Internet connection.

“The core of our mission is to give material to people who need it,” Mr. Khan said. “You could ask, ‘Why should it be free?’ But why shouldn’t it be free?”

For now, Mr. Khan’s small team is subsidized by more than $16.5 million from technology donors, including Bill Gates, Google, the Silicon Valley Community Foundation and the O’Sullivan Foundation. He intends to raise an endowment. And this summer, starting in the Bay Area, where he is based, he plans to hold an educational summer camp.

It is too early to know whether the Khan Academy software makes a real difference in learning. A limited study with students in Oakland, Calif., this year found that children who had fallen behind in math caught up equally well if they used the software or were tutored in small groups. The research firm SRI International is working on an evaluation of the software in the classroom.

Mr. Khan’s critics say that his model is really a return to rote learning under a high-tech facade, and that it would be far better to help children puzzle through a concept than drill it into their heads.

“Instead of showing our students a better lecture, let’s get them doing something better than lecture,” Frank Noschese, a high school physics teacher in Cross River, N.Y., wrote on his blog in June.

But in education circles, Mr. Khan’s efforts have captured imaginations and spawned imitators. Two Stanford professors have drawn on his model to offer a free online artificial intelligence class. Thirty-four thousand people are now taking the course, and many more have signed up. Stanford Medical School, which allows its students to take lectures online if they want, summoned Mr. Khan to help its faculty spice up their presentations.

And a New York-based luxury real estate company credited Mr. Khan with inspiring its profit-making venture: the Floating University, a set of online courses taught by academic superstars, repackaged and sold to Ivy League colleges and eventually to anyone who wants to pay for them.

“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.”

Matt Richtel contributed reporting.

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