What makes a “good” counterexample? Criteria for the comparison of counterexamples and for fostering subject matter-oriented reflections among preservice teachers

In mathematics, counterexamples play a central role in sharpening conjectures and refuting false statements. At the same time, there is an educational dimension, for example, when counterexamples are used to stimulate processes of conceptual change in secondary students. In practice, the answer to the question of whether a specific counterexample will succeed in convincing students of the incorrectness of a statement or the inadequacy of an idea will depend on a variety of factors. We collected criteria which can be used to compare counterexamples in educational settings through an analysis of existing literature, to which we added further, original criteria. In an empirical part, we investigated which criteria preservice teachers (PSTs) would use to compare counterexamples, and whether and how they could make use of criteria introduced by us. We found that across the PST discussions, all previously identified criteria prompted fruitful reflections upon counterexamples, and that comparing counterexamples in this way naturally led to in-depth discussions of the subject matter side of teaching decisions. PSTs discussed multiple criteria which we had not considered, seeming especially interested in context- and learner-dependent aspects of teaching. Thinking in terms of criteria showed potential as a theoretical lens, and for teacher training.