Investigation of the proof characteristics of different mathematical communities: Perceptions and practices

The purpose of this research is to examine the characteristics of mathematicians, mathematics educators, and their undergraduate students regarding their perceptions and practices related to mathematical proof. In this research, the qualitative research method of multiple case study was employed. A total of 34 participants took part in the research, including mathematicians, mathematics educators, and their final-year undergraduate students from a state university in Turkey. Two different data collection instruments were utilized in the research. The first involved semi-structured interviews aimed at understanding participants' perceptions of mathematical proof, while the second employed task-based interview protocols to comprehend participants' approaches (practices) in attempting to prove a mathematical claim. Qualitative content analysis was employed with the blended approach in terms of inductive and deductive manners in research data analysis. Research findings revealed five distinct aspects of participants' perceptions related to mathematical proof. These aspects include proof descriptions, features of proof, tension between school mathematics and university mathematics, proof pathways (nature of their attempts), and the value (importance) assigned to proof. Among the research findings regarding perceptions reveal that mathematicians mainly provide mathematical proof descriptions in terms of a mathematical world-based, while mathematics educators primarily offer teaching-focused descriptions of proof. In terms of students from these two communities, it is evident that the students of the first group mainly present tool-focused proof descriptions, while those of the second group mainly offer process-oriented proof descriptions. Among the research findings regarding practices, it is observed that participants generally shape their proof attempts through both natural and formal routes, and specifically, their approaches diversify by driving on each other in these routes. Particularly, mathematicians are noted for enacting their research through paths that either remain entirely formal or start with some natural ways and later transition to a formal route. However, the other participant groups frequently pursue routes related to natural ways. KEYWORDS: Mathematical Proof, Perception and Practice, Natural and Formal Route, Mathematical Community, Multiple Case Study