How do open-ended problems promote indirect argumentation? Conceptual replications in a Japanese secondary school

The possibility of connecting spontaneous indirect argumentation to indirect mathematical proof has been investigated for decades. It may be effective to use open-ended problems based on the notion of cognitive unity to promote indirect argumentation. Moreover, it also appears useful to analyze students’ indirect argumentation through a model based on the logical structure of indirect proof. However, several convincing critiques of these proposals exist. This study aimed to resolve this dispute and obtain a deeper understanding of indirect argumentation in the process. To achieve this, conceptual replications of previous research were conducted at a Japanese secondary school. The results demonstrated that the exploration of various cases in the situation of an open-ended problem could promote indirect argumentation. Furthermore, the findings indicate that indirect argumentation exhibits diverse characteristics that can be omitted if the analysis is conducted only from a logical perspective.