I was discussing Paul Tough's book with my wife yesterday at breakfast, and we somehow got onto my pet peeve: the mechanical way in which math is taught in schools, and how math gets reduced to arithmetic and counting. Of course the definitive word on this is Paul Lockhart's A Mathematician's Lament.
I looked over the floor of the living room, strewn with random bits of deadly adult-foot-stabbing Lego, and had a minor epiphany. As anyone with small children knows, LEGO sells a lot more Hollywood tie-in kits now compared to the "bags of bricks" that it used to sell. You can buy the Millenium Falcon, an X-wing fighter, a spiderman action scene, or some kit related any cartoon on Nickelodeon.
Which is a pain. Lots of specialized parts, key pieces that if misplaced causes massive bouts of tears and anguished searching, and so on.
But how do kids play with them ? They put it together carefully from the instructions the first time. This lasts about a day or so. Then they get bored, take it apart and mix up all the pieces with the OTHER kits they have, and spend many happy days building all kinds of weird contraptions.
Here's the thing about the kits. They show you how to build fairly complicated devices and doohickeys. The kids pick up on this, and they mix and match the doohickeys in their own new constructions. Essentially they figure out the functions of the different parts, and realize how to build brand new things with them.
But suppose they were told that all they could do was repeatedly assemble the kits (and maybe even different kits) over and over again. It's unlikely they'd learn anything more than just how to assemble kits really quickly. They'd also get terribly bored. In fact, the popular toys in our house are multipurpose objects: any single-purpose toy gets discarded rather quickly.
To cut a long story short, it feels to me that a lot of math (and theoryCS) education at the school and college level involves looking at various "kits", and seeing how they get put together. You get rewarded for putting kits together correctly, but not for playing with the kits and creating bizarre new objects. But finding a way to let students (in an unstructured way) play with mathematical concepts is the key to having them really understand the concepts and their different facets.