I was asked recently by a wise young teacher about the idea of relegating so much of the practice of problem solving in pbl to “homework” time that is outside of class. In many teachers’ mind this takes the observation of the practice of problem solving out of the classroom. I’d love to discuss this a bit here, as I think it might clarify a lot about the procedures of learning problem solving through pbl that I believe have not bee articulated (at least that I haven’t really found articulated in any meaningful way in research articles).

When a student is assigned a problem that they have never seen before and take it home and grapple with it, what do we expect to see the next day? What were they supposed to do that evening? These expectations need to be clear to the student in order for pbl to be successful in terms of learning on many levels. First, a student must be able to dig into their “toolbox” or “toolkit” as many math teachers call it and find remnants and memories that strike a chord with that problem. Others might call this “recall of prior knowledge” or “making connections” or “transfer of skills”, whatever you want to call it, it is something that we want to make happen where a student can find a piece of a problem or solution that have seen before and connect it with a new problem that may not know the solution to in the present. Not only do we want them to connect the idea of a solution, but we want them to connect the idea of the conept as well. Now, do we expect this all to happen in one night of grappling with a new problem for a ninth grade geometry student? Most likely not, that would be practically impossible for a student of average caliber to come up with some mathematically significant new material on their own in one evening. However, if the foundational pieces are laid for them in a way with problems that are stepping stones and arrows pointing in the right direction, it does seem that with the right motivation, they might be able to put down enough recalled information to have a substantial amount of ideas to have productive discussion. This allows the classroom the next day to be the microcosm of the problem-solving real-world of the(substitute appropriate career path here…engineer, business manager, etc.) So in reality, although some of the independent problem solving happens during out-of-class time, a great deal of both the independent and group problem solving practice happens right there in class during the discussion and resolution of the problem the next day.

When a student is presenting the work they have come up with from the previous night’s attempts at a problem, she is presenting the problem solving that happened outside of class. However, even within that presentation she is problem solving. Perhaps she realized her error and because of a comment a classmate made, or an insight shared, or even a prodding by the instructor, she has self-corrected and changed her presentation entirely. This risk-taking and on-her-feet thinking is practice in problem-solving in and of itself. The rest of the class is practicing problem solving because they are learning what they did incorrectly in their recall, how to find the right ways to tell their classmate what she did incorrectly, and also how to be fair and reasonable in their comments to others. The group dynamic in this situation is very important.

So, do I have a problem relegating so much problem solving to students’ work alone for “homework” – no, not really. It does take a lot of coaxing and talking to them about the expectations of the teacher and changing their idea of success on homework. Being explicit about what you value on their nightly efforts is very important and having a grading rubric that does that for them helps – as I do. However, I do believe that having a pbl curriculum sends the message to students that the major goal of the mathematics classroom is that they should be learning to be problem solvers, and that that should be happening everywhere, all the time.