There's something in education you may or may not have heard of—the student-centered approach. Here's what some people think it looks like:
Students doing whatever they want, however they want, as long as it has some tenuous connection to the subject at hand. There are no wrong answers. Math facts are left by the wayside.
Basically chaos, with very little education happening.
I imagine some teachers actually carry it out that way, but that's not the philosophy as I understand it. When we hear the term "student-centered," I think we tend to have ideas of, "Let the student lead the way. Let the student determine everything." So I try not to think about student-centered without including something else.
I envision students out at sea, paddling around in the water, exploring to their hearts' content. The earlier illustration would end there, but when I think of it, each student also has a tether. How much rope they have might vary, but all the lines are connected to a stable post. They'll reach that post from different sides and at different rates—of course, as the teacher, I'm there giving gentle tugs to each rope to urge them in my general direction—but they'll all get to the same endpoint.
That destination is the core principle, the major mathematical idea that's the reason we're doing the activity or exploration at all. Students are empowered to delve into the thick of it, really engage their brains to make sense out of a situation. They see the different approaches their classmates took and discuss whether they're equally valid.
Most importantly, they come away with an understanding of that root concept.
Like most things, easier said than done. Even if I set up a great lesson, it can be awfully tempting to forget those "gentle tugs" and just haul each student in by their tether. It's also easy to run out of time before they reach the destination, and then forget the next day that I've left them adrift.
Before we even get started, then, I'd better make sure they all have life preservers. In other words, setting up a classroom environment where they know the expectations and what to do when they're left without the mathematical understanding we're looking for.
Can you tell the first day of school is coming up?
Wish me luck.