| Students' difficulties in reasoning with logical implication and mathematical proving have been documented widely (Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009). Review of the educational and cognitive science studies of students' reasoning with logical implications and mathematical proving have revealed that their lack of cognizance of counterexamples might be a crucial factor. This study examined the role of logic training and counterexample in enhancing students' logical reasoning and various aspects of mathematical proving, namely, Proof Construction, Proof Validation and Knowledge of Proof Method. In particular, the study hypothesized that logic training emphasizing counterexamples was better able to improve students' reasoning of logical implications as well as mathematical proving, in comparison to the other two approaches emphasizing rule violations and truth tables.;Using a pretest-intervention-posttest experimental design (3 conditions by 2 test trials), students' written and interview data (N = 60) were collected from three Singapore school sites, each over a four-day contact period (including the pretest and posttest administration days). Experimental results showed that logic training emphasizing counterexamples was significantly more effective in improving students' logical reasoning of implication than the other two approaches (p = .0007, large effect size). However, logic training was only similarly effective or ineffective in improving some aspects of students' mathematical proving across conditions.;Interview findings from 12 selected students' works on a new proving task conjectured that students improved their use of deductive inferences in all aspects of mathematical proving after logic training. Moreover, their successes in constructing mathematical proofs were also subjected to two conjectured factors, students' interpretation of implication and mathematical knowledge. These findings suggested the importance of logic training and counterexamples in mathematics education and pointed to further inquiry about the role of students' interpretation of implications and mathematical knowledge in mathematical reasoning and proving. |
