Very interesting explanation. I think you must be familiar with DiSessa's p-prims which ties in with your findings. I remember Steven Pinker's give an account of how Harvard students describe the path of a ball which is rotating about a point and is released, as curved - cited from memory.

On the mathematical front, I would say that Khan Academy renders a great service and renders it free of charge. However, most solutions give rise to new problems. My great fear is that we will have a bunch of people who know the procedures of mathematics which of course free software on the Net can do very well so there is some sort of pointless competition there. The more important component is that students should have a mathematical mindset and should be able to model solutions to real world problems rather than competing with the computer to solve the equations that result from the modelling process. The computer does the mathematics of computation more reliably and of course more rapidly.

Our work with children in Indian schools has shown that some fundamental issues are significant and do not appear to be addressed by the Khan Academy. For instance, students that I have worked with do not appear to have any understanding about the need for symbols/number. Grade 9 and 10 students have indicated that while they can acquire this understanding over the years, the realization that this had been a problem is something they only became aware of after our demonstration but it held them back in prior grades.

Our work has been informed to a great extent by the excellent research conducted in the United States and freely distributed to the world at large. One can only salute the largesse of a people who have given freely to the world massively enabling tools such as Internet, GPS and now moocs - and of course, the Khan Academy. Long may they prosper.