This is the continuation of my report on last weekend’s Asilomar conference. (Previous installment.)
Avery Pickford’s session was about student-created problems. You can read a summary on this blog (Without Geometry, Life is Pointless). Creating problems is, after all, what mathematicians do. Yes, they sometimes explore questions that have been posed by others, but even then, much of the time, their work is a constant back-and-forth between posing and solving problems.
In most classrooms, including mine, there is very little problem-posing by the students. You can think of this as a spectrum where at one end we see the traditional approach where students never see a problem. They are told how to carry out a certain procedure, and then practice that in silence. Somewhere in the middle, are classrooms like mine, where problem-solving is a core feature of learning, but the problems are largely chosen by the teacher or the textbook. Avery is at the other end: he often practices intentional vagueness in the statement of a problem in order to trigger problem-posing by the students. (There are other uses for deliberate vagueness: for example, Dan Meyer uses it to generate student responses to the question: what would we need to know in order to solve a given problem?)
Another technique of Avery’s is to present a situation, or some interesting images, and see what questions arise. As a participant in his workshop, I found this extraordinarily fun and engaging. I loved coming up with my own questions, but given my decades of work as a curriculum developer, the main thing I learned was that there were many legitimate and interesting ways to take off from the prompt, and while I am “good at this”, I should not live under the illusion that I can always find the best problem, or even that there is a best problem. This is not about the student guessing at what the teacher might be thinking. It is genuinely about student thinking and creativity. Many outcomes are possible, including especially unexpected ones.
In fact, one of Avery’s points was that students and groups of students will often gravitate towards formulating problems that make sense to them, problems that will challenge them at an appropriate level. This diversity of problems makes sharing more genuine, as each group will be presenting their own problem to the class. This is so much better than groups sharing similar answers to the same problem, which is not only boring, but also a waste of everyone’s time.
Of course, the teacher does not abdicate his or her role: choosing the prompt and selecting how constrained to make it involves some serious reflection about the usefulness of the problem at a given juncture in a course. Ideally, you want to choose problems that will (a) further your teaching goals, and (b) strengthen students’ ability to to solve similar, though not identical problems. (Avery calls this sideways scalability.) Moreover, as students work the teacher needs to offer useful hints and suggestions, such as “How could we change this problem to make it a little easier?” or “Now that you answered your original question, can you think of other questions we might ask?”, and so on. The teacher also needs to help students keep their focus on the math — there are many interesting questions one can pose, but not all of them are mathematical.
I will discuss some questions this presentation triggered for me in my next post.