Promoting effective mathematical practices in students: Insights from problem solving research

Mathematicians and mathematics educators have been curious about the processes and attributes of problem solving for over 50 years. As mathematics teachers at any level of education, we want to know what teaching practices we can employ to help our students develop effective problem solving abilities. This curiosity has led to numerous investigations of the attributes and processes of problem solving. In this chapter, we describe insights from a study we conducted of the mathematical practices of 12 research mathematicians. We believe these insights are useful to teachers striving to promote mathematical practices in students at all levels–from first-grade mathematics to beginning algebra, calculus, and abstract algebra. Our chapter begins by inviting you to work a problem that our research study posed to 12 mathematicians and to reflect, as they did, on your own problem solving behavior as you attempt to solve this problem. In inviting you to work this problem, our intent is to raise your awareness of the processes, emotions, knowledge, heuristics, and reasoning patterns that you use when working a novel problem. Our research suggests that by reflecting on our own mathematical practices, instructors can become more attentive to the development of problem solving attributes in students (Bloom, 2004). This exercise should make the remaining sections of our chapter more meaningful. In particular, it is our hope that our description of the Multidimensional Problem Solving Framework is more accessible.