CONTROVERSIAL REASONING LEVELS AMONG PROSPECTIVE MATHEMATICS TEACHERS IN REAL ANALYSIS PROOF CONSTRUCTION: A QUALITATIVE CASE STUDY

Abstract

Constructing proofs in Real Analysis requires navigating complex cognitive conflicts, yet the mechanisms that bridge intuitive beliefs and formal logic remain underexplored. This study investigates the dynamics of controversial reasoning among prospective mathematics teachers as they construct proofs of the limits of real-valued sequences. Employing a qualitative case study at a university in Kediri, data were collected from 40 undergraduate students through controversial problem tasks and semi-structured interviews. Data analysis followed the Miles and Huberman framework, comprising data reduction, data display, and conclusion drawing, tracing cognitive trajectories across three levels: initial, exploration, and clarification. Results reveal that reasoning at the initial level is constrained by intuitive misconceptions, in which students recognise contradictions but offer conceptually irrelevant justifications. The exploration level emerges as a critical yet fragmented stage in which students initiate formal strategies, such as mathematical induction, but fail to integrate essential prerequisites for convergence, leading to biased conclusions. The clarification level is characterised by the ability to reconstruct logical consistency through the appropriate use of proof by contradiction. The distinct cognitive patterns observed across the three levels indicate that controversial reasoning constitutes a dynamic, evolving process rather than a static classification, as evidenced by qualitative differences in students' argumentation structures and proof strategies at each level.

Keywords: Cognitive Conflict; Contradiction; Controversial Reasoning; Mathematical Proof