How does problem based learning work anyway?  According to Wikipedia, "Problem-based learning (PBL) is a student-centered instructional strategy in which students collaboratively solve problems and reflect on their experiences."  To me this means, choose problems which will reflect your curriculum and which students want to solve.

Implementation of this in mathematics can be tricky for some topics, even contrived.  If you find yourself really stretching to make a particular concept or unit fit PBL, don’t use it, use some other strategy instead.  However for almost all topics finding a real-life problem, which the students think is interesting or at least has application in their life, is relatively easy.  This is a chance for we mathematics teachers to stretch our creative muscles but it is really important that the problem chosen is either something the students have a direct interest in, or something that they can see someone in their society needing to solve.

The model I use for PBL is this; I describe a problem that exists in our world and needs solving on a regular basis, and I give the students a starting place for solving the problem, then I guide the students through the solution (giving different amounts of advice depending on the understanding of the students). At the beginning of the year the problems are quite regimented, by the end of a school year some students can solve problems mostly unguided.

My objective is to choose problems which ideally weave many different areas of mathematics (or other subjects) into the problem itself.  For example, students were given an assignment to try and decide what the best possible choice of cell phone plan is in the Metro-Vancouver area.  The solution to this involved using linear functions to model the individual cell phone plans, graphical analysis of those linear models to try and determine the best plan for any given number of minutes, algebra to determine the exact number of minutes when different plans intersect, and of course, lots and lots of research into different cell phone plans.

As the students progress through the problem, I can feed them some ideas on how to proceed.  Different groups require different amounts of guidance, and the final product the students produce can vary greatly within a class.  I often find I teach a bunch of related skills to a problem at the beginning of a class, then let the students find the connections and decide how to use the skills during their project.  Most of the time if a given skill is useful, the students find a way to incorporate the use of that skill unto their solution of the problem.

Many of the authentic learning experiences I described in an earlier post can be turned into problem based learning.  You can review these projects and then think of ways you can find problems of your own to use.