I think you're right regarding Khan Academy not necessarily being in opposition to Papert's Constructionism. But I think Papert would argue that learning math is very much like learning programming. In fact, he created the Logo programming language as some sort of "mathland" where learning math would be like learning French in France ( http://www.wired.com/geekdad/2007/03/the_origins_of_/, http://j.mp/h6BiOC)

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.'"

http://www.maa.org/devlin/LockhartsLament.pdf