Learning math is not like learning programming. Math has a quantitative evaluation process.
The kind of scenarios you are talking about both have a quantitative and a qualitative evaluation process.
I.e. you can be creative in your programming and the "answers" but not so much in math.
Also I don't think Kahn Academy is in any opposition to Seymore Paperts "Mindstorms". In fact you could imagine the videos being used to teach children to get started before they go on to hack together their little programs.
What they have done is for the first time made a system that quantifies the learning process so that it benefits the student rather than the school.
Pretty significant if you ask me.
Real mathematicians have often lamented on the approach to mathematics that schools have taken. For example, this essay titled "A Mathematicians Lament" starts like this: "A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. [...] Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the 'language of music.' It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school. As for the primary and secondary schools, their mission is to train students to use this language— to jiggle symbols around according to a fixed set of rules: 'Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.'"
Consider complex numbers. You usually first meet them with imaginary numbers and polynomial equations. But they are extremely obtuse. But if you look at operations on them in terms of vectors, lateral extension of the reals to make the picture in terms of solvable polynomial equations complete, or in terms of polar coordinates, rotations, trigonometry, and e you start to get a fuller picture. But if you really want to get them you program a simple system (scaling and rotating triangles for example) that allows you to leverage all those properties. You build in the identities and operations. And then you experiment. Play with it. Play is vital to learning. With that, you get motivation and a fuller understanding than a mere that is just the way it is. With that deeper more intuitive understanding you are able to remember and internalize it better. Making the future topics easier but more importantly, developing the ability to draw connections rather than merely leaving each concept in isolation.
That is why maths is so hard. So much of it is completely divorced to how they were originated and completely unmotivated and magical. With no intuition future learning is further crippled, building apprehension via a negative feedback loop.
Again I am a big proponent of Seymores work but that is not the point here.
The point is that Kahn Academy allows you to learn math in your own tempo and that is the future of education. That doesn't mean that other things wont be important too but to say that it's not the future of education is simply missing the greater point IMHO.
People who have never been able to understand it before now suddenly do. We don't all need to be math wizards.
I also think, especially for learning kids maths, not to underestimate how powerful learning as playing is in motivating concepts. I do think such a thing done right, where you would replace a bunch of videos with one game would be even better than the videos as the future of education. As that type of active learning allows not just variable pacing but also variable exploration and creativity. And by building a repertoire of stories for concepts, it counters how alien and not straight forwardly relatable to real life experience more abstract maths is. A fact that makes learning maths hard.
As a first test of this idea I hope to someday make or see a game where kids program little battling bots and in the process learn concepts from high level physics and maths. as a side-effect. without realizing they are learning. I guess based on personal experiences, I feel there is still a class of kids that the khan academy style won't reach.
That kids learn concepts from high level physics and maths does not mean they learn formal math.
I am not saying that they need to learn formal math (in fact there are plenty of arguments against spending so much time trying to teach kids any math)
But it's important to understand that if everyone have to learn concepts in their own way they will loos the ability to communicate with each other.
The ability to transfer metaphors from something we understand into the world of Mindstorms and through that develop conceptual understanding of various areas does not necessarily transcend into understanding of math.
It might though create something else and quite wonderful. But formal math it probably wont be.
Likewise with math (and many other disciplines), there is a core set of skills and patterns that need to be mastered - once you truly understand those skills and patterns - you can then use them in creative ways to solve the problems.
Khan's great skill is in helping to explain those skills and patterns, leaving the teacher time to help the student master the practice of them.
That's where hacker culture comes from. People talking to each other, helping each other.
While I understand your concern in general, it seems unfounded in this case.
The week after that, he went over all the concepts again, but seeing the problem, then seeing the way to solve it seems better, to me.
First, I totally buy the argument that a YouTube video can be a better method of delivering content than a book or even, as Kahn says, a live teacher. However, learning is much more than that. Learning is about imagining, playing, creating, sharing and reflecting (Resnick 2007, PDF: http://j.mp/gqnc6f).
To say that learning math is about watching a video and drilling on some problem set, is like saying that learning Python is about doing a tutorial and solving some exercises. Hackers know this well. You don't really learn a programming language until you build some project, especially a personally meaningful one. Creating something not only helps you learn the language but also its community, culture, practices, how to ask questions and get feedback on your work.
Think about it, would you rather hire a developer that has finished some tutorials or one that has actually hacked some projects? The same logic applies to most other fields. Artists and scientists know this as well: you learn by doing in the social context of a community.
Unfortunately, schooling has taken the "hacking" away from learning and has focused on improving content delivery, artificial problem-solving and test-taking. Often using technology. I liked when he mentioned that the teachers have more time to mentor students. However, their mentoring is most likely still focused on the artificial and decontextualized approach of traditional schooling.
Initiatives like the Khan Academy will probably play a role in the future but to really improve education the system and the approaches should change. One should look at exemplar learning environments like the hacker communities or even the Brazilian Samba schools. Settings that are "real, socially cohesive and where experts and novices are learning" (Papert 1980, Google Books: http://j.mp/dLNbGG).
In 1971, Ivan Illich outlined an inspirational alternative educational environment where "the child grows up in a world of things, surrounded by people who serve as models for skills and values. He finds peers who challenge him to argue, to compete, to cooperate, and to understand; and if the child is lucky, he is exposed to confrontation or criticism by an experienced elder who really cares."
Illich said that the four resources learners need are "things, models, peers, and elders". In his 1971 vision, Illich said that "the operation of a peer-matching network would be simple. The user would identify himself by name and address and describe the activity for which he sought a peer. A computer would send him back the names and addresses of all those who had inserted the same description. It is amazing that such a simple utility has never been used on a broad scale for publicly valued activity"(http://j.mp/hNUVV8). Sounds familiar? When I read Illich, I think of HN, StackExchange, and the larger ecology of spaces that hackers have created around social learning. My hope is that these and other similar approaches scale up and get adopted more broadly along with resources like the Khan Academy, OCW and others.