Mathematical problem solving: an evolving research and practice domain

Research programs in mathematical problem solving have evolved with the development and availability of computational tools. I review and discuss research programs that have influenced and shaped the development of mathematical education in Mexico and elsewhere. An overarching principle that distinguishes the problem solving approach to develop and learn mathematics is to conceptualize the discipline as a set of dilemmas or problems that need to be explored and solved in terms of mathematical resources and strategies. In this context, relevant questions that help structure and organize this paper include: What does it mean to learn mathematics in terms of problem solving? To what extent do research programs in problem solving orient curricular proposals? What types of instructional scenarios promote the students’ development of mathematical thinking based on problem solving? What type of reasoning do students develop as a result of using distinct computational tools in mathematical problem solving?     

Musing on the use of dynamic software and mathematics epistemology

Different computational tools may offer teachers and studentsdistinct opportunities in representing, exploring and solvingmathematical tasks. In this context, we illustrate that theuse of dynamic software (Cabri Geometry) helped high schoolteachers to think of and represent a particular task dynamically.In this process, the teachers had the opportunity of identifying,exploring and supporting mathematical relations that emergedduring the solution of the task. We distinguish problem-solvingepisodes that teachers exhibited while understanding and representingthe task, thinking of a solution plan, searching and presentingmathematical arguments and looking for connections.     

Problem solving and the use of digital technologies within the Mathematical Working Space framework

ZDM – Mathematics Education - The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological...     

High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies

The aim of this study was to characterize and discuss ways of reasoning that prospective high school mathematics teachers develop and exhibit in a problem-solving scenario that involves the coordinated use of digital technologies. A conceptual framework that includes Virtual Learning Spaces (VLS) and Resources, Activities, Support and Evaluation (RASE) essentials is used to introduce and support a problem-solving approach to structure learners’ problem-solving activities that encouraged them to share ideas, discuss and extend mathematical discussions beyond formal settings. Main results indicated that prospective high school teachers relied on a set of tool affordances (dragging objects, looking and exploring object’s loci, using sliders, quantifying and visualizing mathematical relations, etc.) to formulate, explore and identify properties or relations to share, discuss and support mathematical conjectures. In this context, the participants recognized and valued the importance of using several tools to both dynamically represent and explore mathematical tasks and to share and constantly refine their mathematical ideas and problem-solving approaches.     

The use of digital technology in finding multiple paths to solve and extend an equilateral triangle task

Mathematical tasks are crucial elements for teachers to orient, foster and assess students’ processes to comprehend and develop mathematical knowledge. During the process of working and solving a task, searching for or discussing multiple solution paths becomes a powerful strategy for students to engage in mathematical thinking. A simple task that involves the construction of an equilateral triangle is used to present and discuss multiple solution approaches that rely on a variety of concepts and ways of reasoning. To this end, the use of a Dynamic Geometry System (GeoGebra) became instrumental in constructing and exploring dynamic models of the task. These model explorations provided a means to generate novel mathematical results.     

On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.

Editor: Uri Wilensky

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