Introduction to International Perspectives on Problem Solving Research in Mathematics Education

Abstract

Any field of research and innovation must be exposed to revisions, criticisms and to an intense scrutiny not only to discuss the state of the art but, hopefully, to identify prospective changes and new areas of study and exploration as well.Problem Solving has been such an area, with a prominent place in mathematics education and whose contributions continuously appear in conference proceedings, handbooks, journals, books and, more recently, in digital endeavors. Problem Solving involves an approach that fosters reflection and delving into mathematical ideas to explain individuals' cognitive behaviors within social media. Here, we argue that ideas do not live by themselves isolated from the semantic networks that sustain the life of cognition: meaning. These networks constitute a key ingredient for developing understanding and structural perspective of concepts through problems. In the long term, (and maybe not that long) networks provide integration of knowledge that learners need to construct and integrate in order to gain a wide perspective.Problem Solving drives developments through research programs, curriculum design, teachers' mathematical education, and mathematical instruction at the level of the classroom. Taking this and more into due account, and in order to identify current trends in mathematical problem solving and to foster a further exchange of ideas within our community, we discussed the present project with Bharath Sriraman, the editor of The Mathematics Enthusiast, who generously accepted to devote a special issue of the Journal to Mathematical Problem Solving. We invited colleagues, who have made significant contributions to this field, to contribute to the special issue. Previously, we had identified some lines of development to eventually frame their contributions. Of course this was not meant as a restriction to their freedom; rather to orient their possible directions. Some questions were posed and discussed to help us identify possible themes to consider in the volume. Thus, we tried to answer: What are the current trends in problem solving research and what are the main results that influence teachers ' practices and curricula design? In addition, with the significant development of digital tools and environments, we are in need to understand to what extent the present research agenda is being or need to be transformed by their influence. This touches, for instance, deep epistemic issues concerning the nature of valid mathematical reasoning and results in mathematics in the classroom. We have to take into account that mediation tools are not neutral, neither from the cognitive nor from the epistemic viewpoint. That the knowledge students generate and/or appropriate, is intertwined with tools. However, we cannot forget that a school culture always leaves significant marks on students' and teachers' values. Artigue (2005, p. 246) states that these [previous] values were established, through history, in environments poor in technology, and they have only slowly come to terms with the evolution of mathematical practice linked to techno- logical evolution. Thus, the school culture requires the gradual re-orientation of its practices to gain access to new habits of mind and to the new environments resulting from a serious presence of digital technologies.We consider that how we understand the learning of mathematics through a problemsolving approach is deeply related, today, with the presence of the mediation tools that students will find and use in and outside the classroom.We have shared with the invited authors a list of themes; we would like to mention some of them we consider particularly relevant.Mathematical Problem Solving Foundations. Any domain of study needs to make explicit tenets and principles that support and justify its academic agenda. As we mentioned before, we are interested in documenting the extent to which theoretical and pragmatic frameworks are helpful to explain the problem solvers' development of new mathematical knowledge. …