Ahh, and herein lies the rub…practice or not to practice – that is the question. Do you feed into the tradition of repetition that perpetuates students’ perceived notions of success in learning mathematics? Do you continue to lead them to believe that if they can do 25 of the same type of problem that they are now successful problem solvers? Where does our responsibility lie in being mathematics educators – to the standardized tests to which our students must submit or to the larger skills that will move them forward in all aspects of their lives? Or can there be a balance?

We read in newspapers almost monthly about school districts that have initiated a “progressive” mathematics program with all good intentions to foster creativity and problem solving, higher order thinking skills. But the article inevitably is not about the good that is being done, but the outcry of angry parents or unsatisfied board members is usually the point of the article and how we all need to just “go back to basics” (see NY Times Education section article by Tamar Lewin, As Math Scores Lag, a New Push for the Basics, November 14, 2006 or Samuel Freedman’s ‘Innovative’ Math, but Can You Count? November 9, 2005). Do we just ignore these issues and pound ahead as if only our agenda matters?

We’ve had to find a balance this year in our geometry curriculum that allows the students who need it, the practice to gain certain important skills. We would make the decision to do practice only when we all felt that not only the majority of students needed the practice, but we also saw the skill as a necessary tool to move onto their next course. Let me see if I can give an example. We spent a great deal of time between October and December having students write equations of medians, perpendicular bisectors and altitudes. Although we don’t feel that the ideas of the points of concurrency is important for further study, we definitely felt it was worth it for them to have the practice in order to move onto Algebra II/Trigonometry next year. We also practiced factoring and solving quadratics too.

However, we are going on the assumption that it is not the specific content that we cover that is important, but the practicing of problem solving skills through the content itself. From my experience and from readings of research, I believe that the process of solving problems can be practiced and with that important repetition of the risk-taking and potentially being wrong, it becomes much more easy to solve problems simply because you are open to trying. I’ve seen this in over 16 years of teaching, that students don’t want to try because they don’t want to be wrong. Their definition of success in problem solving is one that is based on always using a method that a teacher has shown them and getting the right answer the teacher tells them is right. This is what routine practice gets you, students who can match a teacher’s method, not necessarily students who can be independent problem solvers.

So, I think that the right type of practice can be an extremely helpful friend to PBL. As long as what the students are practicing is a balance between necessary skills and independence in problem solving. Both types will move students forward in their confidence and ability to take risks.