We math teachers love to talk about problem solving as a desirable curricular goal. However, I find that many of us don’t really agree as to what constitutes a problem. Not an original thought here, but in my mind I make distinctions between exercises and problems. I see it this way – an exercise is any challenge in front of you where the path to a solution is clear. It might be a technique you’ve been practicing, a new skill that has just been introduced, a specific formula to be applied. You might not get the answer correct, but it’s not because you don’t know WHAT to do. A problem is a challenge where the path to a solution is unclear. It might involve tying together multiple strands, creating (discovering?) a new technique that has yet to be presented, it may involve reaching across curricular boundaries to call on skills from other courses. So, if I am interested in teaching problem-solving, I need to have my cherubs on board and agree with me about what a problem is. I was happy to see the following link (http://fcit.usf.edu/math/resource/fcat/strat.htm) in Fawn Nguyen’s most recent post. Proud to see my old home state has a fairly cogent presentation of problem-solving. Of course, the father of talking about teaching problem-solving is still Georg Polya (http://teacher.scholastic.com/lessonrepro/lessonplans/steppro.htm) A MAJOR goal of mine next year is to reach consensus with my students about this job of ours.
Interesting post, I agree that we need to decide what we mean by problem solving. How are you going to communicate this with your students?
I’ve been thinking about this for a while Simon and one of the ways I intend to reach some consensus with my students is through my assessments. I have always felt that the way a teacher assesses tells the student, in the most concrete terms, what is valued in a classroom. I intend to take some of my old tests and share them with my students in advance to talk about which of the questions qualify as exercises versus which ones are seen as problems. I also intend to have certain assessments announced in advance as skills/exercise tests while some are to be announced as problem tests. To help reduce stress, the skills assessments will likely carry more weight or at least be more numerous. This feels like a baby step but I think I can gain some traction.
I hope you’ll keep us posted on how this communication with students goes. I learned about Polya’s distinction between an exercise and a problem a few years ago in a course taught by Wil Reimer at Fresno Pacific University. Wil may have been the first person to make it onto my “list of people I want to be like when I grow up” (at least in a math education sense; as a kid I thought it would be pretty sweet to be like Michael Jordan).
I feel that as I’ve strengthened, clarified, and streamlined some aspects of my classroom and teaching, I’ve drifted away from regularly providing students genuine problem solving opportunities. The motivation has definitely been rekindled, so it’s time for me to do some of what you describe above. I completely agree about the messages we send to students (through our assessments) about what is truly valuable. I need to some more thinking about what that looks like in my classroom and in our department, but thanks for the push. 🙂