Pre-service Elementary Teachers’ Conceptions of Counterexamples

Abstract

This study aimed at investigating two main issues related to counterexample construction: the appropriateness of counterexamples and the types of arguments that are often used when refuting a false conjecture.  Twelve pre-service elementary teachers who demonstrated a wide range of reasoning skills participated in this study. The data revealed various phenomena among pre-service teachers’ conceptions of refutations. While all participants who demonstrated deductive reasoning were aware of the fact that one counterexample was sufficient to refute a false statement, the majority of the participants who had not yet developed deductive reasoning failed to recognize that one counterexample would be enough and/or they tend to believe that providing more than one counterexample would support the argument further to refute the statement. Furthermore, the participants from the deductive proof scheme attempted to construct a general refutation or to provide a justification for their constructed counterexamples while the participants from different proof schemes only provided specific counterexamples without further explanation or justification. This study also documented various misconceptions that PSTs held regarding to the underlying concepts in which they were being asked to refute—the concepts of area and perimeter or quadrilaterals—and how their misconceptions affected their constructed counterexamples or decisions as to whether the presented statements were true or false.