Reformulating: Approaching Mathematical Problem Solving as Inquiry

The Gestalt psychologist Karl Duncker (1903-1940) characterized problem solving as getting from where you are to where you want to be through successive reformulations of the problem until it becomes something you can manage. That view can be seen in a recent European project to promote inquiry as a means of learning mathematics and attracting students to its study. It can also be seen in increased research efforts to study problem formulation in mathematics. Considerations of how to educate teachers of mathematics to approach problem solving as inquiry should include attention to questions of metacognition and to the cognitive demand of a problem-in much the same manner as George Polya (1887-1985) promoted such attention. For Duncker and Polya, we solve problems by replacing our unsuccessful efforts by successful ones through a heuristic inquiry process.