Thoughts About Research On Mathematical Problem- Solving Instruction

Abstract

Abstract: In this article, the author, who has written extensively about mathematical problem over the past 40 years, discusses some of his current thinking about the nature of problem-solving and its relation to other forms of mathematical activity. He also suggests several proficiencies teachers should acquire in order for them to be successful in helping students become better problem solvers and presents a framework for research on problem-solving instruction. He closes the article with a list of principles about problem-solving instruction that have emerged since the early 1970s.Keywords: mathematical activity, problem solving, problem-solving instruction, proficiencies for teaching, craft knowledge, research design, as a craft, teacher planning, metacognition..IntroductionMy interest in problem as an area of study within mathematics education began more than 40 years ago as I was beginning to think seriously about a topic for my doctoral dissertation. Since that time, my interest in and enthusiasm for problem solving, in particular problem-solving instruction, has not waned but some of my thinking about it has changed considerably. In this article I share some of my current thinking about a variety of ideas associated with this complex and elusive area of study, giving special attention to problem-solving instruction. To be sure, in this article I will not provide much elaboration on these ideas and careful readers may be put off by such a cursory discussion. My hope is that some readers will be stimulated by my ideas to think a bit differently about how mathematical problem solving, and in particular problem-solving instruction, might be studied.Setting the stageMost mathematics educators agree that the development of students' problemsolving abilities is a primary objective of instruction and how this goal is to be reached involves consideration by the teacher of a wide range of factors and decisions. For example, teachers must decide on the problems and problem-solving experiences to use, when to give problem particular attention, how much guidance to give students, and how to assess students' progress. Furthermore, there is the issue of whether problem is intended as the end result of instruction or the means through which mathematical concepts, processes, and procedures are learned. Or, to put it another way, should teachers adopt teaching for problem solving, -an ends approach-or teaching via problem solving -a means approach?1 (I say more about means and ends later in this article.) In my view, the answer to this question is that both approaches have merit; problem should be both an end result of learning mathematics and the means through which mathematics is learned (DiMatteo & Lester, 2010; Stein, Boaler, & Silver, 2003). Whichever approach is adopted, or if some combination of approaches is used, research is needed that focuses on the factors that influence student learning. Unfortunately, as far as I know, no prolonged, in-depth, programmatic research of this sort has been undertaken and, as a result, the accumulation of knowledge has been very slow. Moreover, the present intense interest in research on teachers' knowledge and proficiencies demands that future problem-solving research pay close attention to the mathematical and pedagogical knowledge and proficiencies a teacher should possess (cf., Ball, Thames, & Phelps, 2008; Hill, Sleep, Lewis, & Ball, 2007; Moreira & David, 2008; Zazkis & Leikin, 2010).But before discussing problem-solving instruction, let me first say a few things about mathematical problem solving. This short discussion will highlight how my thinking has changed about the nature of problem and other forms of mathematical activity.Some claims about Problem SolvingAmong the many issues and questions associated with problem-solving instruction I have worried about during my career, several have endured over time. …