"Khan Academy may be one of the most dangerous phenomena in education today," declared a recent blog posting by Karim Ani, the founder of Mathalicious, a site that offers richly integrated math videos and curricula.

Ani’s comments triggered an outpouring of reactions from mathematics teachers around the U.S. A couple of days later, Dan Meyer, a former math teacher now working on his Ph.D. at Stanford, weighed in with his own attack—this less about Khan Academy per se and more about Silicon Valley entrepreneurs’ approach to capturing math in a computer program.

"The medium defines, changes, and distorts the message," he wrote. "When you attempt to distribute mathematics through any of these media [such as YouTube videos, digital photos, MP3s, PDFs, blog posts, spoken words, and printed text] it changes the definition of mathematics.

"Silicon Valley’s entrepreneurs, venture capitalists, and big thinkers assume a shared definition of 'mathematics.' They innovate around the delivery of that mathematics. CK-12 has PDFs. Khan Academy has YouTube videos. Apple has iPad apps. ALEKS and Junyo have computer adaptive tests. Very few of them understand that each of those delivery media changes the definition of mathematics.

"Even worse, at this moment in history, computers are not a natural working medium for mathematics (PDF)."

Some 150 or so comments later—along with another hundred or so comments on HackerNews—the tensions between the teacher-centric and the entrepreneur-centric viewpoint continue to emerge.

Some teachers have been stung by the notion that Khan—a guy in his pajamas who doesn’t seem to give a wit about decades of pedagogical research—is teaching online. Others are distressed that when he gives away his lessons for free, no one else will be able to make a living selling their more carefully constructed curricula. (Mathalicious’s Ani has a foot in this camp.)

Still others (including both Ani and Meyer) grind their teeth at Khan’s here’s-how-you-solve-it approach to math. "This paint-by-numbers method of instruction emphasizes procedures—how to do math—but ignores the conceptual understanding that’s central to authentic learning: what math means. At its core, this is a function of ineffective instruction, which to a large degree is related to ineffective content," writes Ani.

Meyer takes the argument a step further, contending that most edtech tools take a "just solve it" approach to math that is fundamentally incompatible with a more experiential approach. He writes: "The argument that these innovations are all supplements to one another, and all of them to a teacher, and that none are meant as a complete solution is a common one in [Silicon Valley]. But I think it is a mistake to assume that the different definitions of mathematics promoted by each of these different innovations will all harmonize with one another."

Part of the confusion stems from uncertainty about the purpose of education. Most people would agree we don’t simply want children to succeed at taking tests. (Even on this point, however, some contend that test scores— such as 8th grade math scores—are a good predictor of later success, notes Harvard’s Roland Fryer).

Most of us also want students to be creative, curious, and ready to learn new skills no matter what direction their lives take them. That doesn’t happen in every classroom: "The reality is we only need Khan style edu to outrun the 50% of teachers who are just horrible (so we can increase class size), scare 40% of them to work harder using Khan in the new bigger classes, and reward the last 10% with sizable pay raises," contends Morgan Warstler on Meyer’s blog.

At the very least, educators (and parents) hope students will understand how to reapply their newly acquired knowledge to broader mathematical situations. That is a tricky skill to teach, however. On Meyer’s blog, teacher David Wees wrote:

"What I find frustrating is that the over-emphasis on the value of the data [Khan has] collected. I used the Khan Academy with my students (mixed in with a problem solving and project based approach) at the beginning of this year, and within 6 weeks I abandoned it (after two assessments I’d given to students external to the program) because I found no relationship between what my weakest students had 'mastered’ according to the types of questions the Khan Academy provides, and the students ability to use their calculations in any other context. They could do 30 problems in a row flawless related to the rules of exponents, for example, but not solve any problem directly related to the rules of exponents that they hadn’t seen on the Khan Academy videos or exercises. The transferability of what my weakest had learned was extremely low. By contrast, the approach absolutely worked fine for my stronger mathematics students, who were able not only to transfer what they had learned, but to work at a pace I could not hope to match."

Finally at the core of the debate is a deep tension that all Internet denizens should recognize: Who’s in charge? Should educators, deeply versed in pedagogical research and countless years of feet-in-the-classroom, be calling the shots? Who gave the newcomers, typically entrepreneurs, many whom last set foot in class when they were students, the right to frame education?

Other industries have grappled with this tension, too, as the Internet became the dominant platform. In industries where "experience" mattered more than degrees or credentials, the newcomers have rewritten the rules. Travel agents, for instance, have seen their profession redefined by the likes of Travelocity and Kayak. Traditional journalistic organizations have similarly absorbed a body blow as the influence and credibility of bloggers has grown.

But professions where credentials are essential have changed more slowly: Lawyers still handle most legal affairs. Although online medical sites have flourished, few patients have severed their relationship with a flesh-and-blood doctor because they preferred the likes of Drkoop.net.

So what about teachers? Are they more like credentialed doctors who should have a commanding say over what tools are introduced into their practices or more like journalists, forced to prove their value every day? Or can teachers themselves learn to work with the entrepreneurial community to deliver curricula that have rich pedagogical roots? From the vantage of the millions who are Khan fans, that middle ground looks compelling:

"I have watched more than 700 videos on Khan Academy, over the past 11 months, and to me the single most significant impact of the videos is that it inspires me to learn more," wrote one commentator on Meyer’s blog.

"Can’t you work with Khan Academy, and the teachers trying to implement it in class rooms, to 'convey complementary definitions of mathematics’? Can’t you use the excitement generated by Khan Academy to raise the standard of teaching and learning in classrooms, and elsewhere?"