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David Heckman, a mathematician, was accustomed to lecturing to the class, but he had to take on the role of a roving mentor, responding to raised hands and coaching students when they got stumped

Like institutions at every level of American education, it is going through some wrenching changes. The university has lost 50 percent of its state funding over the past five years.

alarmingly high numbers of students showing up on campus unprepared to do college-level work.

Let’s start with what Khan Academy is. Khan Academy is a collection of video lectures that give demonstrations of mechanical processes. When it comes to this purpose, KA videos are, on the average, pretty good. Sal Khan is the main reason; he is approachable and has a knack for making mechanical processes seem understandable. Of course, his videos are not perfect. He tends to ramble a lot and get sidetracked; he doesn’t use visuals as effectively as he could; he’s often sloppy and sometimes downright wrong with his math; and he sometimes omits topics from his subjects that really need to be there (LU decomposition in linear algebra, for example). But on balance, KA is a great resource for the niche in which it was designed to work: giving demonstrations of mechanical processes.

But let’s also be honest about what Khan Academy is not. Khan Academy is not a substitute for an actual course of study in mathematics. It is not a substitute for a live teacher. And it is not a coherent curriculum of study that engages students at all the cognitive levels at which they need to be engaged. It’s OK that it’s not these things. We don’t walk into a Mexican restaurant and fault it for not serving spaghetti. I don’t fault Khan Academy for not being a complete educational resource, because it wasn’t designed for that purpose. Again, Khan Academy is a great resource for the niche in which it was designed to work. But when you try to extend it out of that niche — as Bill Gates and others would very much like to do — all kinds of things go wrong.

When we say that someone has “learned” a subject, we typically mean that they have shown evidence of mastery not only of basic cognitive processes like factual recall and working mechanical exercises but also higher-level tasks like applying concepts to new problems and judging between two equivalent concepts. A student learning calculus, for instance, needs to demonstrate that s/he can do things like take derivatives of polynomials and use the Chain Rule. But if this is all they can demonstrate, then it’s stretching it to say that the student has “learned calculus”, because calculus is a lot more than just executing mechanical processes correctly and quickly. To say that it is not — that knowledge of calculus consists in the ability to perform algorithmic processes quickly and accurately — is to adopt an impoverished definition of the subject that renders a great intellectual pursuit into a collection of party tricks.

Salman Khan talks about how and why he created the remarkable Khan Academy, a carefully structured series of educational videos offering complete curricula in math and, now, other subjects. He shows the power of interactive exercises, and calls for teachers to consider flipping the traditional classroom script -- give students video lectures to watch at home, and do "homework" in the classroom with the teacher available to help.