I'd like to see more exploration of how new concepts in mathematics were discovered and why.
The teaching on the Khan Academy is superb (ALEKS, I'm not so sure about). The site is also incredibly well-organized. I think that the people who run it could turn it into a magnificent piece of work that would survive for a long time if they added in-depth discussions of these kinds of questions:
1. How has mathematics developed over time?
An example would be the appearance of zero in different places in the historical record, the invention of variables and systems used before Descartes started using x and y, the use of negative numbers, etc. Why did zero pop in and out? What finally cemented its use?
2. What has hindered mathematical development?
An example here would be resistance to the use of negative numbers, which were seen as preposterous until relatively recently. Complex/imaginary numbers had the same problem. IMO, probing the reasons for resistance to new ideas is important not only to advancing mathematics, but also for understanding scientific and technical progress as a whole.
3. What unanswered questions or needs drove the development of new mathematical techniques?
An example here would be calculating the volume of a pyramid and a frustum (a pyramid with its top gone). The ancient Egyptians worked out a way to do the very non-trivial frustum calculation without calculus (it's a standard topic in many or most calculus classes today). What are theories about how they did that? Learning about how other people succeeded when you would think they wouldn't have been able to can teach a lot about how to succeed today when confronted with an apparently impossible problem.
IMO, we teach our students how to calculate, but we do a poor job of explaining how to solve fundamental problems. When I say this, I don't mean that we don't teach practical applications like "you can use trig to calculate the height of the cliff." I mean that we need to be teaching patterns of thinking that can help drive major new discoveries. Without this, we can keep calculating (as Richard Feynman advised), but we'll have a lot of trouble making big conceptual leaps. This stuff is part of learning mathematics, not some kind of philosophical tangent, and it's lacking overall today. The KA could fill a big void here.
Sure, very highly motivated people can figure this stuff out for themselves over the course of years or decades, but wouldn't it be, you know, if nothing else, more efficient to put it together up front? This is the kind of stuff that makes an educated population or, looking at it more narrowly, it's something gifted students would eat up.
