Commentary: Linking epistemology and semio-cognitive modeling in visualization

To situate the contributions of these research articles on visualization as an epistemological learning tool, we have employed mathematical, cognitive and functional criteria. Mathematical criteria refer to mathematical content, or more precisely the areas to which they belong: whole numbers (numeracy), algebra, calculus and geometry. They lead us to characterize the “tools” of visualization according to the number of dimensions of the diagrams used in experiments. From a cognitive point of view, visualization should not be confused with a visualization “tool,” which is often called “diagram” and is in fact a semiotic production. To understand how visualization springs from any diagram, we must resort to the notion of figural unity. It results methodologically in the two following criteria and questions: (1) In a given diagram, what are the figural units recognized by the students? (2) What are the mathematically relevant figural units that pupils should recognize? The analysis of difficulties of visualization in mathematical learning and the value of “tools” of visualization depend on the gap between the observations for these two questions. Visualization meets functions that can be quite different from both a cognitive and epistemological point of view. It can fulfill a help function by materializing mathematical relations or transformations in pictures or movements. This function is essential in the early numerical activities in which case the used diagrams are specifically iconic representations. Visualization can also fulfill a heuristic function for solving problems in which case the used diagrams such as graphs and geometrical figures are intrinsically mathematical and are used for the modeling of real problems. Most of the papers in this special issue concern the tools of visualization for whole numbers, their properties, and calculation algorithms. They show the semiotic complexity of classical diagrams assumed as obvious to students. In teaching experiments or case studies they explore new ways to introduce them and make use by students. But they lie within frameworks of a conceptual construction of numbers and meaning of calculation algorithms, which lead to underestimating the importance of the cognitive process specific to mathematical activity. The other papers concern the tools of mathematical visualization at higher levels of teaching. They are based on very simple tasks that develop the ability to see 3D objects by touch of 2D objects or use visual data to reason. They remain short of the crucial problem of graphs and geometrical figures as tools of visualization, or they go beyond that with their presupposition of students' ability to coordinate them with another register of semiotic representation, verbal or algebraic.