Mathematical visualization, manipulatives and geometric problem solving: a case of study

The role of mathematical visualization in problem solving process and the incidence of material tools on apprehension of mathematical concepts have been subjects of intensive research for a long time. In the present paper we analyze the use of a manipulative in solving a space Geometry problem, providing a critical account of its theoretical framework and its effectiveness in various teaching contexts. Previous researches (Leikin, Stylianou, & Silver, 2005) examined students’ use of visual representations, across various ages and mathematical education levels, proposing a challenging task to the study participants: describe the net of a truncated right cylinder. Given the visual nature of the problem, any solution strategy required a translation between 3D and 2D representations, regardless of the more or less advanced mathematical knowledge. In the behavior of the successful visualizers, similarities were identified, such as selection of what needs to be visualized and transfer of visual images in a “symbol system”, see Nemirovsky (1994); while both young and older low-achieving students reported difficulties in isolating object’s relevant characteristics. About the same problem, a physical manipulative was designed (Cumino, Spreafico, & Zich, 2017) in a dissemination context where Geometry was a tool for understanding architectural shapes, under the need of quickly and correctly communicating the solution, avoiding mathematical formalizations: It shows a right cylinder truncated by oblique planes of various inclination and allows to obtain its net by a simple unwrap. Currently, this manipulative is being tested at the university level, on first year students of the bachelor program in Architecture and at the middle school level, on second year students of a Technology course; in both teaching contexts, different for age and levels of mathematical knowledge, it is proving useful in making students able to elaborate individual solving strategies, to set up the problem correctly and to improve its translation in symbolic language for the holders of advanced mathematical tools, speeding up teachers’ interventions.