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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.

the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.

Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.

A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching “pure” math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations.

What's the difference between a 10-inch skillet and a 12-inch skillet?

the rate of evaporation of a liquid is proportional to its exposed surface area, a sauce in a larger, flatter skillet will reduce far more rapidly than one in a taller, narrower pan

It is amazing what was accomplished when I removed my assumptions of what a “test” must be.

Richard Ludlow started the nonprofit Academic Earth two years ago after M.I.T.'s OpenCourseWare helped him pass linear algebra as a Yale undergraduate. His site offers the courses of 10 elite universities — 130 full courses and more than 3,500 video lectures. Viewers can turn the tables on professors and grade courses. Other guidance includes "Editor's Picks" and "Playlists," lectures selected around a theme like "First Day of Freshman Year" and "You Are What You Eat."

Connexions, started at Rice University 10 years ago, debundles education for the D.I.Y. learner. Anyone can write a "module," the term for instructional material that can be a single sentence or 1,000 pages. Connexions hosts more than 16,000 modules that make up almost 1,000 "collections." A collection might be, say, an algebra textbook or statistics course.

Daniel Colman is a curator of sorts. He sifts through the vast amount of free courses, movies and books offered online to find what he considers the very best in content and production value. Then he features them on Open Culture, the Web site he founded in 2006. It's a task in keeping with his mission as associate dean and director of Stanford's continuing education program.