Reflections on Problem Solving Theory and Practice

Abstract: In this article, the author reflects on the current state of mathematical problem solving, both in theory and in instruction. The impact of the book Mathematical Problem solving (Schoenfeld, 1985) is also discussed, along with implications of problem solving today with the advent of 21st century technologies.Keywords: Mathematical problem solving; Mathematics teaching; Mathematical learningIntroductionMy book Mathematical Problem Solving (Schoenfeld, 1985), which I shall refer to as MPS) was published more than 25 years ago. MPS, which was fundamentally concerned with research and theory, had been developed in dialectic with a course in problem solving at the university level. The book provided a theoretical rationale for the course, and evidence that it worked; the course was an existence proof that, with the "right" kinds of instruction, students could become more effective problem solvers. The book-plus-course addressed a series of theoretical and pragmatic questions, some of which they answered, some of which they suggested answers to, and some of which they left unaddressed. Either directly or by logical extension the ideas in the book had the potential for significant curricular impact, if the "lessons" in them were taken seriously.The question is, what has been the fate of the ideas that the book and the course embodied? Which ideas survived, which flourished? Which evolved in unpredictable ways, which withered with unfulfilled promise? I am grateful to the editors for the opportunity to reflect on the past and to think about future opportunities.I begin by describing what, in my opinion, were the achievements, failures, and potential of that early work (which, of course, built upon and reflected the state of the field in 1985). This is followed by a characterization some of the main outcomes of the evolution of problem solving research and development. There is, of course, a huge literature on problem solving. It is impossible to do justice to that literature, and my comments will be selective. My most general comments are based, in part, on the volume Problem solving around the world - Summing up the state of the art (Torner, Schoenfeld, & Reiss, 2008). That volume provides a recent overview of theory and practice (and to some degree, curricular politics) in a wide variety of nations. This article will update my article in that volume (Schoenfeld, 2008), characterizing recent and potentially significant events in the U.S.Problem Solving as of 1985 - a retrospective viewIn theoretical terms, what MPS offered in 1985 was a framework for the analysis of the success of failure problem solving attempts, in mathematics and hypothetically in all problem solving domains. "Problem solving" at its most general was defined as trying to achieve some outcome, when there was no known method (for the individual trying to achieve that outcome) to achieve it. That is, complexity or difficulty alone did not make a task a problem; solving a system of 100 linear equations in 100 unknowns without the use of technology might be a real challenge for me, but it is not a problem in the sense that I know how to go about getting an answer, even if it might take me a very long time and I agonize over the computations.The core theoretical argument in MPS, elaborated slightly in Schoenfeld (1992), was that the following four categories of problem solving activity are necessary and sufficient for the analysis of the success or failure of someone's problem solving attempt:a) The individual's knowledge;b) The individual's use of problem solving strategies, known as heuristic strategies;c) The individual's monitoring and self-regulation (an aspect of metacognition); andd) The individual's belief systems (about him- or herself, about mathematics, about problem solving) and their origins in the students' mathematical experiences.Regarding (a), little needs to be said; one's mathematical knowledge is clearly a major determiner of one's mathematical success or failure. …