First, I think that Khan Academy gets a bit of a bad wrap. I'm pretty sure it was not Sal Khan's intent to "revolutionize" or "reinvent" education by posting his videos online. In my opinion, a public misconception about mathematics, learning, and education has allowed KA to rise to it's current position. The public views mathematics as a body of factual information, one that is passed from the "knower" to the "student." Learning, then, is perceived as memorizing/understanding factual information. This public perception applies to much of education in general, but especially to mathematics education.
Recently, the MTT3K video exploited some mistakes in Khan's explanation of multiplying positive and negative numbers. Among other criticisms, the makers of the video point out that Khan mistakenly refers to the "transitive property" and explains "-4 x 3" as "negative fours times itself three times." In addition, the video points out some questionable pedagogical decisions by Khan which might be confusing or difficult for students first learning the topic and that fail to explain why the multiplication facts are true.
I welcome the dialogue that is happening surrounding these videos, but want to take the discussion of why Khan Academy is an ineffective learning tool one step further. In doing so, I want to reference Constance Kamii and her application of Piagetian principles to teaching:
"Piaget's theory of memory is very different from the empiricist belief that 'facts' are 'stored' and 'retrieved.' According to Piaget, a fact is 'read' differently from reality by children at different levels of development because each child interprets it by assimilating it into the knowledge he has already constructed."
In this next quote, she refers to facts about addition. I think the same could also easily be said for multiplication:
"In Piaget's theory, there is no such thing as an 'addition fact.' A fact is empirically observable. Physical and social knowledge involves facts but not logico-mathematical knowledge. The fact that a ball bounces when it is dropped is observable (physical knowledge). The fact that a ball is not appreciated in the living room is also observable (social knowledge). But logico-mathematical knowledge consists of relationships, which are not observable. Although four balls are observable, the 'four-ness' is not. When we add 4 to 2, we are putting into an additive relationship two numerical quantities that each of us constructed, by reflective abstraction; 4 +2 equals 6 is a relationship, not a fact."
At first, these quotes seem hard to swallow. However, Piaget (and others) have conducted many experiments to show that even children who "knew" facts about addition lacked the ability to perform on different tasks involving those same operations.
We construct in a way that is personal to us and relative to our current ways of understanding. My point in all of this is that, video or no video, you can't make someone learn. The 'knowing' that is so prized in education comes from every person's natural ability to think and from the human tendency to maintain internal/cognitive equilibrium. My suggestion is that we don't tell students what they should know or how they should think. Give them a task that pushes on their way of knowing, let them do mathematics, and watch and listen closely as they sort things out together.