Aarón Reyes-Rodríguez

Teachers' Use of Computational Tools to Construct and Explore Dynamic Mathematical Models.

To what extent does the use of computational tools offer teachers the possibility of constructing dynamic models to identify and explore diverse mathematical relations? What ways of reasoning or thinking about the problems emerge during the model construction process that involves the use of the tools? These research questions guided the development of a study that led us to document the process exhibited by high school teachers to model mathematical situations dynamically. In particular, there is evidence that the use of computational tools helped them identify and explore a set of mathematical relations or conjectures that appear throughout the interaction with the task. Thus, the participants had the opportunity of fostering an inquisitive approach to the process of model construction that led them to formulate conjectures or mathematical relations and to search for ways to support them.

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.

Connecting Dynamic Representations of Simple Mathematical Objects with the Construction and Exploration of Conic Sections.

Different technological artefacts may offer distinct opportunities for students to develop resources and strategies to formulate, comprehend and solve mathematical problems. In particular, the use of dynamic software becomes relevant to assemble geometric configurations that may help students reconstruct and examine mathematical relationships. In this process, students have the opportunity of formulating questions, making conjectures, presenting arguments and communicating results. This article illustrates that simple geometric configurations can be used to generate all conic sections studied in a regular course of analytic geometry. Observing and interpreting loci, searching for mathematical arguments (including empirical reasoning) and presenting results are problem-solving activities that seem to be enhanced with the use of dynamic software. During the development of the activities, students can exhibit a line of thinking in which they constantly reflect on the meaning and connections among mathematical concepts.

Using a dynamic geometry system to integrate analytic and synthetic knowledge in the solution of geometry problems

In this study, we documented how the systematic use of a dynamic geometry system DGS during problem solving became a means of integrating synthetic and analytic concepts of geometric knowledge. We analysed solution paths for tasks of geometric construction, developed by participants of a problem-solving seminar during two sessions and identified the limitations of purely analytic approaches and the advantages of integrating synthetic and analytic techniques. The results indicate that solving problems with the support offered by a DGS increases the opportunities that problem solvers have to interpret algebraic procedures from a geometric perspective and to construct meaning of mathematical concepts.

Designing Technology-Based Tasks for Enhancing Mathematical Understanding Through Problem Solving

In this paper, we propose some organizing principles that can be useful for high schools or bachelor mathematics teachers when designing technology-based instructional tasks. It is widely accepted that tasks are the most important aspect to promote students’ mathematical understanding, since tasks offer opportunities to attain relevant sensorial experiences fostering the construction of mental images as sources of meanings for mathematical concepts. In this vein, we reflect on the work developed by three bachelor mathematics teachers who participated in a problem solving seminar. The main points identified during task design involved recognizing how mathematical concepts are structured around the task, and which are needed to approach it, and proposing a hypothetical learning trajectory in which technology plays a role as amplifier and reorganizer of cognitive processes.

Integrating Synthetic and Analytical Aspects of Geometry Through Solving Problems Using a DGS

In this paper we document and analyze the extent to which the systematic use of a Dynamic Geometry System (DGS) in problem solving activities can become a means to integrating the synthetic and analytical aspects of geometry. We analyze some solution paths for a task of geometric construction, implemented by three participants at a problem-solving seminar. We identify limitations of purely analytical approaches as well as the usefulness of integrating analytical and synthetic techniques to construct or justify a solution path. We observed that solving problems of geometric construction without the support offered by digital technology reduces the opportunities that a solver has to interpret algebraic procedures from a geometric perspective and to construct meaning for mathematical ideas and concepts.