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ChristianKl comments on Interesting critique of British education by outgoing advisor (warning: some politics) - Less Wrong
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That accepts basic premises about education that I don't think make sense. I don't think that there any basis to believe that video explaining something are a time effective way of learning something. The same goes for straight lecturing of information.
Children don't learn their native language because their parents explain them how it works.
On the level of making education policy the easist thing would be to simple get rid of the curriculum and let every school teach what they consider to make sense. Additionally you facillitate lots of knowledge exchange between teachers.
That would allow innovation. It might even allow for children spending more time in front of Khan Academy.
That reminds me of Aikido grandmaster Koichi Tohei who made the point that the really important skill is to teach people who teach teachers effectively.
Even if video teaching is (today?) not the best way, I think it would be nice to create a feedback loop, because feedback is what's missing in education now. Once you have a system "here is a video, students see it, students take tests", you can experiment with various changes and see whether those changes improved the results. The same thing could be done with books, of course. The important part is to allow the education to replicate, and measure its results. Then a way to gradual improvement (instead of random drift) is opened.
These days, education is typically done like this: random people create new educational theories mostly based on pseudoscience, teachers are taught these theories, teachers do random things in classrooms, nobody really evaluates what's going on as long as nothing extreme happens. -- In this situation if you ask questions like "what are the different known ways to teach integrals, and how efficient is each of them", no one really knows, because no one ever measured that in any meaningful way. The only answer you could get is the latest fashion in pseudoscience, for example "you should support multiple learning styles and, uhm, make it funny", which, even if you'd happen to agree with it, is not specific enough to give measurable results.
It is certainly not the only way to teach, but it is a way that could be measured. We should at least try it experimentally, so see what kinds of results it could produce.
And measure the outcomes, and reward those who have the best ones. The rewards can be financial, but also the prestige. (If you have a private school, and the state makes everyone known that you are the best school in the country, you are going to get a lot of money even without the state giving it to you.)
Going more meta! But the question of measuring is still here.
That's the huge advantage of online learning. Performance and behavioral metrics can be as detailed as you like.
That leaves the question about which outcomes you measure. I think it's okay to have a world with some school that run like KIPP where there a lot of measurement and others that run like Sudbury Valley with has feedback principle like internal elections and reviews how many of it's student succeed at college.
When it comes to the specific example of teaching integration I think that will be done best via some computer tool.
I guess that 20 hours of time investment into practing a well developed Anki deck on integration would leave most students with more knowledge of integration afterwards.