Teaching Mathematics Differently?

http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-about-math-that-hold-americans-back/281303/

... article seems to support the idea that traditional and formalistic approaches to mathematics were themselves an unnecessary dressing. If true, an interesting idea: one that has resonance with the notion that meaning is more important than analysis and that "getting it" is something different from calculating. Reminds me of how Wittgenstein thought that the commutative law for multiplication could simply be taught with a picture of dots. Hence (a x b) = ( b x a) could be shown thusly: