There’s this trope, mostly appearing in martial arts fiction, of a learning process where tasks that appear menial and pointless are actual lessons that impose the skills or knowledge needed without the student being aware that they’re learning anything.
My favorite of these is in The Matador series by Steve Perry1. In it, neophytes try to follow an Arthur Murray style sequence of footsteps. Each successive step is not only more difficult than the previous, but it build and expands on the skills needed in all of the previous steps.
In my dreams, this is how I could teach math: A sequence of problems, carefully constructed, such that each problem provides a small measure of growth from the previous problem – so close that no instruction or direction is needed, but enough of an advancement that the student needs to make a cognitive leap to get there.
I realize this is a pipe dream, for several reasons. Students learn at different rates and with different styles, some need a lot more reinforcement of concepts than others do – there is no one ideal path.
Furthermore, a lot of what we teach in math is not the enlightenment of the concepts, but conveniences of notation. There is no real intuitive way to grasp those – we as teachers need to impart those conveniences onto their existing understanding.
Nevertheless, I believe there is great power in providing just the right problem to a student. Rather than having to convince and cajole a student into knowledge, a well designed question can shake up their assumptions, and provide the spark of inquisitiveness that so many teachers wish their students had.
Key to developing these questions is an understanding of how students develop theories and understandings, and how they can develop incomplete or flawed theories that nevertheless function adequately for a significant subset of the problems they face2.
I’m hoping to keep this as an ongoing subthread in this blog, and I hope that it may actually someday develop into a fleshed out philosophy that is concrete enough to share and defend.
1 I think I just outed how horrible my taste in recreational literature is. I have no apologies – crappy SciFI is a staple of the geek world.
2 cf Polya.