Characterizing Levels of Reasoning in Graph Theory

Abstract

This work provides a characterization of the learning of graph theory through the lens of the van Hiele model. For this purpose, we perform a theoretical analysis structured through the processes of reasoning that students activate when solving graph theory problems: recognition, use and formulation of definitions, classification, and proof. We thus obtain four levels of reasoning: an initial level of visual character in which students perceive graphs as a whole; a second level, analytical in nature in which students distinguish parts and properties of graphs; a pre-formal level in which students can interrelate properties; and a formal level in which graphs are handled as abstract mathematical objects. Our results, which are supported by a review of the literature on the teaching and learning of graph theory, might be very helpful to design efficient data collection instruments for empirical studies aiming to analyze students' thinking in this field of mathematics.

Proof levels of graph theory students under the lens of the Van Hiele model

“…The purpose of this work is to provide empirical support for the validity of the levels of proof in graph theory through the lens of the Van Hiele model proposed by González et al (2021). This implies the search for evidence of the indicators provided by these authors for the levels, as well as the study of their adequacy to the particularities of the Van Hiele model.…”

“…We can now describe the main indicators of the Van Hiele levels for graph theory (González et al, 2021). It is easy to see that they have the same nature as the Van Hiele levels for geometry:…”

“…Moreover, they are able to understand formal proofs and even reproduce a few logical steps but they cannot produce themselves formal proofs, which characterizes level 4 students. González et al (2021) provide a theoretical characterization of students' reasoning in graph theory based on Van Hiele levels. To present this characterisation, we first provide some fundamentals of graph theory for the sake of completeness.…”

“…Furthermore, González et al (2021) organize the descriptors of each level according to the processes proposed by Gutiérrez and Jaime (1998) adapted to the field of graphs. In particular, the evolution of the process of proof in graph theory and in geometry are analogous from levels 2 to 4.…”

“…In addition, this model has proved to be useful to describe the learning of other mathematical topics such as local approximation (Llorens & Pérez-Carreras, 1997), convergence of sequences (Navarro & Pérez-Carreras, 2006), convergence of series (Jaramilllo, 2000) or functions (Nisawa, 2018). Thus, González et al (2021) perform a theoretical analysis that produces a characterization of the learning of graph theory through four levels of reasoning under the lens of the Van Hiele model, whose validity must be tested in empirical studies as the present paper. This characterization is organised through the development of the processes of reasoning that students may activate when dealing with graphs: recognition, use and formulation of definitions, classification, and proof, thus providing descriptors of levels for each process.…”