Journal of Mathematical Behavior 24 (2005) 217–220 Mathematical problem solving: What we know and where we are going Jinfa Cai a,* , Joanna Mamona-Downs b , Keith Weber c a University of Delaware, USA b University of Patras, Greece c Rutgers University, USA Available online 11 October 2005 In the last three decades, there has been a great deal of educational research on mathematical problem solving, and this research has deepened our understanding of problem solving and related pedagogical issues immensely. As we reflect on the research trends in mathematical problem solving, we realize just how dynamic research on mathematical problem solving is. This is hardly surprising when one considers some of the fundamental questions that the field has to address: What is mathematical problem solving? What are the cognitive processes used in solving mathematical problems? What are the purposes of problem solving? What are the actual mechanisms that students use to learn and make sense of mathematics through problem solving? What is the teacher’s role in implementing problem solving in the mathematics classroom? The views of the mathematics education community on each of these questions have evolved over time and are still in flux. It is appropriate to periodically take stock of the field by examining how mathematics educators are currently looking at problem solving and seeing what issues currently have need of further research. One of the purposes of this special issue of the Journal of Mathematical Behavior is to conduct such an examination. It should be noted that while we will refer to a single special issue, it is really two issues combined into one. This special issue originated from the 10th International Congress of Mathematics Education’s Topic Study Group 18: Problem Solving in Mathematics Education. The general aims of the Topic Study Group were to provide a forum for those who are interested in any aspect of problem-solving research at any educational level, to present recent findings, and to exchange ideas. We set up three specific goals for the Problem Solving Topic Study Group: (1) to examine the understanding of the complex cognitive processes involved in problem solving; (2) to explore the actual mechanisms by which students learn and make sense of mathematics through problem solving, and how this can be supported by teachers; and (3) to identify future directions of problem-solving research, including the use of information technology. The Topic Study Group received a good response. There were 22 papers presented, authored by nearly 30 researchers from 13 different countries. A total of roughly 350 mathematics education researchers from around the world attended at least one of the sessions for the Topic Study Group. Most of the papers in this special issue are from those who presented at the Topic Study Group. In addition, we invited a few other researchers to submit papers in order to cover various aspects of problem-solving research that we wished to be represented in this issue. This special issue includes 12 papers, each addressing at least one of the three goals listed above. The first six papers that appear are empirically based; in these papers, the authors present the results of the fieldwork that they have conducted and also raise research questions for future studies. The remaining six * Corresponding author. 0732-3123/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2005.09.014