I read Mathematical Problem Solving by James W. Wilson, Maria L. Fernandez, and Nelda Hadaway at http://jwilson.coe.uga.edu/emt725/PSsyn/PSsyn.html
Overall, I thought this was a good article that emphasized teaching students “how to think” was of primary importance. Unfortunately, in many math classrooms, students are taught “what to do” and this is usually done as a linear process with several steps that leads to a specific answer.
Although the beginning of this paper was foundational and provided needed definitions and frameworks, it also contained a lot of tips on how to make this transformation in the classroom. For example, in Geometry when dealing with algorithms, problem solving can be improved by having the students create their own algorithms.
Polya was quoted many times in this article and supports the importance of looking back when problem solving. This gives the student a great opportunity to learn from the problem. However, this rarely happens. Students tend to stop when getting the answer and teachers want to move on when the student can get the right answer when assessed on a multiple choice test (commonly used in standards assessments).
I particularly liked this summation from the authors: we need to change the definition of mathematics from algebra, geometry, or calculus to exploration inquiry, discovery, plausible reasoning, and problem solving. Problem solving can be fun and this should be used to stimulate students’ interest and enthusiasm.
Unfortunately, much of this has to start with the teacher and he or she will probably have to create assessments towards this end. The article included many examples and they all were typically open ended. There were links where readers could find problems that encouraged students to think. The authors concluded with using hints, Planning/Representation/Doing Boards, and students solving problems verbally out loud as ways to encourage problem solving.