Every day, I give my students problems to solve. When I do that, I'm also telling them:
- "This is a problem that you can make progress on."
- "This is a problem that it is worth your time and effort."
- "This problem is new and not the same as what you've studied before."
Because I teach my lessons in units, there's an additional implication present:
- "This problem is connected to what we studied yesterday."
Are these things that I want to be communicating to my students? I don't know. (Those wiser than me will hopefully chime in on this.) But I do have a few observations to make:
- All of these implications make the problem easier to solve. My assurance that the problem is solvable and that it's worthy of their time helps them take the leap into the problem with confidence. The implication that this question has something to do with what we have been studying significantly narrows the available tools and techniques to choose.
- This is fundamentally different from the way problems are outside of the classroom. When I find a question that I want to make progress on, I have no assurance that it has a good or interesting solution or that I'll be able to do it. I have no idea what tools I'm going to need, and making those decisions is part of the difficulty of the problem.
What do we do about this? What can we do about this? We can try to present problems to students that are more akin to how they're found in the world, but the mere fact that we're offering them invests the problem with all of the implications detailed above. If we want to eliminate the suggestion that a problem is connected to the previous day's ideas we could eliminate units and integrate our topics more densely.
Other than that, to really help students solve problems as they're found out there, we need to create more opportunities for them to find problems on their own.
Not like I do that, but hey, aspirations, right?