The crux of Problem-Based Learning is to elicit the right question from students that you, the teacher, are equipped to answer. This requires the teacher posing just the right problem to elicit just the right question that points to the right standard.
Our existing knowledge and schema determine what we wonder so kids wonder kid questions and math teachers wonder math teacher questions. Sometimes those sets of questions intersect, but they’re often dramatically disjoint.
Which makes Geoff’s “crux” a form of mind control, or maybe inception, which is impossible. Kids wonder so many wonderful and weird things. And even if that practice were possible, I don’t think it’s desirable, since it seems to deny student agency while pretending to grant it. And even if it were desirable, I wouldn’t have the first idea how to help myself or other teachers replicate it.
If PBL is to survive, it needs a different crux. Here are two possibilities, one bloggy and one researchy.
First, Brett Gilland:
[The point of math class is to] generate critical thought and discussion about mathematical schema that exist in the students minds. Draw out the contradictions, draw attention to the gaps in the structures, and you will help students to build sturdier, creatively connected, anti-fragile conceptual schema.
Second, Schwartz & Martin:
Production seems to help people let go of old interpretations and see new structures. We believe this early appreciation of new structure helps set the stage for understanding the explanations of experts and teachers – explanations that often presuppose the learner will transfer in the right interpretations to make sense of what they have to say. Of course, not just any productive experience will achieve this goal. It is important to shape children’s activities to help them discern the relevant mathematical features and to attempt to account these features (2004, p. 134).
Notice all the teacher moves in those last two quotes. They’re possible, desirable, and, importantly, replicable.
2016 Jan 12. Logan Mannix asks if I’m contradicting myself:
As a science teacher follower of your blog, I’m not sure I follow. Isn’t that what you are trying to do with many of your 3 act problems? Get a kid to ask questions like “is there an easier way to do this” or “what information do I need to know to solve this”?
I am interested in question-rich material that elicits lots of unstructured, informal mathematics that I can help students structure and formalize. But I never go into a classroom hoping that students will ask a certain question. I often ask them for their questions and at the end of lesson we’ll try to answer them, but there will come a moment when I pose a productive question.
The possibility of student learning needs to rely on something sturdier than “hope,” is what I’m saying.
2016 Jan 13. Geoff Krall writes a post in response, throwing my beloved Harel back at me. (My Kryptonite!) It’s helpful.