More on numbers and symbols

Torigoe’s work, which I discussed in  Numbers vs. symbols and Numbers vs. symbols again, is beginning to spread through the educator blogosphere.  Dan Goldner, in his blog Work in Pencil, wrote a post Substitute, then simplify, which discusses the insights he has gotten into his own teaching:

It never occurred to me that carrying symbols through a problem is more demanding than substituting numbers into as many variables as possible, as soon as possible.

and

This explains everything! It’s why my precalculus students can’t articulate relationships between abstract quantities. They’ve never had to! So they don’t know how.

At least not yet. The technique of “give one specific problem, then a second, then a general one” and hope they make the leap has not worked. So how can I build the capacity to reason with symbols?

Dan’s question is precisely the correct one.  How does one build the capacity to reason with symbols?  Clearly the standard math and science curriculum is not succeeding at this for most students—only about the top quarter of those taking first-year physics in college at a selective university seem to have acquired the skill (which probably translates to about 5–10% of high-school graduates).

The problem is not one of “learning physics” but of a general math deficiency, and so needs to be addressed in math classes (as well as science classes).  Has anyone used any pedagogical approaches that get students to reason abstractly with symbols proficiently (and not just plug-and-chug)?  Has there been any educational research into this fundamental problem of math education?

I don’t know of any, but I’m no expert in educational research. Anyone got any pointers for me?