A statement about mathematical understanding in the Common Core State Standards was recently pointed out to me:
One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. (CCSS for Mathematics, page 4).
No surprise that FOIL is singled out as the example of school mathematics where procedure trumps understanding. For this particular topic, I think that using generic rectangles to visualize the distributive law is better than relying on mnemonics at all, but I'm sure there are other equally good ways of avoiding the FOIL trap. Generally, whatever the topic, the challenge is to find representations that extend existing understanding rather than applying rules without comprehension.
Posts about generic rectangles are here and here.
