My gut reaction is to say that all three of these ways of thinking (interactive, visual, and symbolic) are all useful and should be used where their advantages

My gut reaction is to say that all three of these ways of thinking (interactive, visual, and symbolic) are all useful and should be used where their advantages can be best applied. As a shallow example, I think that the basic visual/interactive representations can be fantastic teaching tools, and once you get an intuitive feel for how the system behaves, you can then switch to symbolic representation and move on to the more higher-order systems which those building blocks are composed of. I thought it was interesting that when he compressed the paragraph describing an equation to is algebraic representation (x2 + 2x = 10), we immediately grokked the algebraic version better, because we have loads of experience reading and manipulating the algebraic expression. If I think back to my own learning career, however, I learned the basic tenets of algebra using -- you guessed it -- interactive and visual representation through Hands-On Equations [1] [1] http://www.borenson.com/ http://www.borenson.com/ (now that I think about it, Hands-On Equations serves as a very nice "tldr" for this speech and the related concepts)