Another Take on Problem Based Learning for Math

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I have been wrestling with the marriage of Project Based Learning (PBL) and Math for nearly 8 years now.  A few years ago I moved towards Problem Based Learning (PrBL) instead of Project Based for math. It just seemed to make more sense.

With PrBL there are shorter time periods to complete a problem which, to a math teacher, frees up more time for "Math Skills." Of course a veteran math teacher, using inquiry-based teaching (PBL or PrBL fall into this niche), will find opportunities for practicing math skills.

Once a teacher decides to pursue PrBL, he or she will need to figure out which math units fit best. Or at least that is what I used to think. Now, after years of working with this, I realize that instead of figuring out which units and then what specific problems to work with, there is a better way.  Teachers need to start the planning process with great "Open-Ended Questions."

Now I use the term "open-ended" loosely because many of the problems I found with that label were actually closed problems.  The open-ended problems I am looking for may have multiple solutions but they may just have multiple paths that can be taken to get to an appropriate solution.

So here's a problem I found for middle school students (taken from the Math Forum):

I can see this being used as an introduction to area. During the first day's questioning I would hope there would be obvious questions about how to calculate area. But I would also hope to hear questions about whether the "whole number dimensions" have to be in feet. Could they be in "whole" inches? How about a question about whether "rectangular" means it has to be four sided - could it be a composite figure of rectangles? There might be a question that comes up that I would never think of.

The beauty of this is that the students can explore their specific questions to solve the problem going down a path of their choosing. At the end of the week the real question is: "Can the students solve a 4-sided, rectangular figure, with a length of L and a width of W?"

My big AHA! then was this - what if I found 20 or 30 or 50 great open-ended questions that represent content I want my students to learn during a school year?  I could then lay these questions alongside my district Year at a Glance or Pacing Guide to see where the questions would make sense on the calendar.

Now as a teacher gets to each unit they will already have 2 or 3 (or 5) open-ended questions that have been vetted. These will be the problems used in class to foster inquiry.  These problems will take 3 or 4 days from start to finish. There will be opportunities for questioning and brainstorming; creating drawings or other models; presentations; and reflections.

For example, the questioning and brainstorming might be completed on the first day. The modeling (along with the requisite critical thinking) might be on day two. On the third day there would be an opportunity for an informal presentation of how the groups solved their problems, followed by a reflection. On the last day of this PrBL there might be an assessment on the learning that was completed.

This all started from having a sound, content specific, question that fueled inquiry, questioning, and thinking. Using an open-ended question allowed students to go in different directions in pursuit of answers. Successful students ask questions, think critically, and defend their ideas. That is where deeper learning happens and that is how we help our students develop perseverance.