We name things for reference, and hopefully for ease of reference, to draw attention to the thing named. But naming also classifies and hence causes us to look at the named thing in particular ways, the chosen symbol stressing some and ignoring other attributes of the named object. Naming something gives us power over it, particularly in algebra, as we can transform and combine expressions involving the unknown – to find out more about it (p. 127).
This is the strongest case for algebra. Your ability to speak, think, and use variable notation makes you powerful – particularly when you interact with computers. But how often do students think of variables in math class and feel powerful? Those experiences aren't simple to devise.
I read Pimm's excellent book over the holiday in preparation for my dissertation proposal. I've pulled out several pages worth of quotes and supplemented them with a) my analysis and b) some details about my upcoming study. Comments are turned on in the Google doc, so let's talk about it.
Perhaps using words as the descriptor ("number of songs") instead of using X (as in "Let x = the number of songs on Dave's iPod") would be a step in the right direction? The same level of rigor without the confusion of what X equals.
How often do students name their own variables?