Exploring Deductive Reasoning Patterns in Proving Logical Statements Using Equivalence Theorem

Deductive reasoning is central to mathematical proof, yet university students frequently struggle to transform logical statements through equivalence theorems. This study analyzed mathematics education students' deductive reasoning abilities and explained the reasoning processes underlying high, moderate, and low performance in formal logical proof tasks. An explanatory sequential mixed-methods design was employed. Quantitative data were collected from 46 mathematics education students through five open-ended deductive reasoning test items, while qualitative data were obtained from six purposively selected students through semi-structured interviews. Quantitative data were analyzed descriptively, and qualitative data were analyzed through data condensation, data display, and conclusion drawing. The results showed that students' overall deductive reasoning performance was relatively low (M = 56.56; SD = 19.83), with 60.87% of students categorized as low performers. Students performed better in identifying propositional forms (68%) than in selecting equivalence theorems (54%), maintaining logical transformation consistency (49%), and producing valid conclusions (44%). Qualitative analysis revealed three reasoning patterns: strategic-reflective reasoning among high performers, partially developed but inconsistent reasoning among moderate performers, and procedural-fragmented reasoning among low performers. These findings indicate that successful logical proof construction depends not only on knowledge of equivalence rules but also on theorem selection, transformation monitoring, and metacognitive regulation. The study contributes to mathematics education by clarifying the cognitive and metacognitive characteristics of deductive reasoning in formal logic tasks and by offering implications for proof-oriented logic instruction.