The ability of students and teachers to use counter-examples to justify mathematical propositions: a pilot study in south Korea and Hong Kong. (English)

ZDM, Int. J. Math. Educ. 45, No. 1, 91-105 (2013).

Summary: Counter-examples, which are a distinct kind of example, have a functional role in inducing logically deductive reasoning skills in the learning process. In this investigation, we compare the ability of students and prospective teachers in South Korea and Hong Kong to use counter-examples to justify mathematical propositions. The results highlight that South Korean students performed better than Hong Kong students at justifying propositions using counter-examples in algebra problems, but both did equally well in geometry problems. In terms of the prospective teachersâ€™ ability to justify propositions using counter-examples in two particular topics, properties of the absolute value function and parallelogram, Hong Kong prospective teachers performed relatively weakly in the absolute value problem but better in the parallelogram problem compared with South Korean prospective teachers. The weaknesses and strengths of students and prospective teachers in generating counter-examples associated with logical reasoning in mathematical contexts in the two regions indicate ways of improving the effectiveness of learning and teaching mathematics through the use of counter-examples.