How visualization and argumentation are articulated in research on teaching and learning geometry

Visualization and argumentation are central processes in the learning of geometry, giving rise to specific lines of research in mathematics education. However, it is worth asking whether they should be developed independently or if it is possible and in what way an articulation between them can be promoted. The objective of this paper is to search for this articulation within the most recent advances in empirical research on the teaching and learning of geometry in Spain. The analysis is approached through the tasks used in research employing two theoretical frameworks which have been developed in Spain, the Onto-Semiotic Approach and Configurational Reasoning. Two main types of task were found in the selected studies: visual tasks and conjecture-and-proof tasks. The results show that in visual tasks the articulation between visualization and argumentation is only necessary in some tasks, in which argumentation aims to justify or to describe the external representation provided as a solution. However, in conjecture-and-proof tasks the articulation between visualization and argumentation must always be present to resolve the task. Analyzing this articulation helps to detect and gain a better understanding of the difficulties that students encounter in solving these types of task. In visual tasks, argumentation helps to understand difficulties associated with the use of visualization. In conjecture-and-proof tasks, visualization determines the argumentation followed, in such a way that many of the difficulties originate more from the visualization than from the process of argumentation.