The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.
It reminded me of the recent article in the Chronicle about Berkinstein and Graff's book.
Both rely on formulas, in a way, but--and here's the important part--they don't begin and end with formulas. The abstractions allow students to apply the concepts but NOT, as detractors fear, to stop at applying principle A to material B in a bad, evil, wicked, formulaic, five-paragraph essay kind of way to make students march in lockstep as prescribed by Satan's minions who want to stifle creativity. Instead, they allow students to conceptualize ideas in new and different ways.
It's like riding a bicycle. If you had to think about how to ride a bicycle every time you rode one, or reconceptualize it every time lest you not have the "true and authentic" experience of discovery, you'd be exhausted and you'd never get anywhere. You'd be so busy learning to ride that you'd never find the pleasure in riding.
Isn't there room for both discovery and actually learning--all right, even memorizing--a few concepts? I honestly don't see the problem with this.