THE TEACHER'S ROLE IN DEVELOPING
STUDENTS' PROBLEM SOLVING
Problem solving is the "heart of mathematics". Successful problem solving requires knowledge of mathematical content, knowledge of problem solving strategies, effective self-mentoring, and a productive disposition to pose and solve problems.
Teaching problem solving requires even more of teachers, since they must be able to foster content knowledge and positive attitudes in their students. A significant portion of a teacher's responsibility consists of planning problems that will give students the opportunity to learn important content through their explorations of the problems, and to learn and practice a wide range of problem solving approaches and strategies.
The teacher must be courageous, for even a well planned lesson can veer off into unchartered territory. Students may make novel suggestions as they try to solve problems; they may make observations that give rise to new conjectures or explorations; they may suggest generalizations whose validity may be unknown to the teacher. Teachers must exercise judgement in deciding what responses to pursue; they must recognize the potential for both productive learning and improved attitudes when students generate new ideas. Teachers must also recognize that not all responses lead to fruitful discussions, and that time constraints do not allow them to pursue every interesting idea. It is the teacher's job to make the tough calls.
To create an environment in which students reflect on their work as they engage it, the teacher must also be reflective. In fact, teaching is itself a problem solving activity. Effective teachers of problem solving must themselves have the knowledge and dispositions of effective problem solvers.