Preservice Secondary Mathematics Teachers’ Conceptions of Counterexample

Reasoning and proof play a key role in secondary geometry curriculum and should be well grasped by students and teachers. As an important aspect of reasoning and proof, disproof by counterexample, can be used as an effective instructional tool in helping students generate conjectures. In my teaching and research of various levels of geometric reasoning and proof for pre-service teachers, I noticed students were challenged by the role and use of counterexamples. Their ability to visualize and create counterexamples was also limited by using paper and pencil. Previous studies have shown that incorporating the dragging feature supported by dynamic geometry environments (DGEs) can promote student understanding and reasoning ability. The goal of the study is to understand the role of technology in unpacking and developing pre-service secondary mathematics teachers’ (PSMTs) conceptions of the definition and application of counterexample. Preliminary results indicate the usefulness of the DGE tool in helping PSMTs understand the meaning and application of counterexample, especially for those who had limited understanding or lower proficiency about the axiomatic system in Euclidean geometry