Addressing the problem of “problem solving”

Abstract “Problem Solving” has been taken to mean many different things. On the one hand, the origin of problem solving in school mathematics must be credited to George Pólya. Pólya’s conception of problem solving encompassed the full breadth of mathematical sense-making, which includes: seeking, finding, and representing patterns; making conjectures and reasoning productively about them; and using a range of techniques to make progress on, and solve, problems that represent non-trivial challenges for the individuals working on them. On the other hand, “problem solving” in school curricula has at times been taken to mean implementing rote procedures to obtain the answers to simple exercises that students have been taught to solve. This essay traces the many meanings of “problem solving” from the mid-20th century to the present and offers the author’s opinion regarding potentially productive curricular and research directions regarding mathematical sense-making, in the spirit of Pólya’s intentions.