Secondary school teachers' awareness of numerical examples as proof

Some mathematical statements can be validated by a supportive example or refuted by a counterexample. Our study investigated secondary school teachers' knowledge of such proofs. Fifty practising secondary school teachers were first asked to validate/refute six elementary number theory statements, then to suggest justifications that students might give for the same statements, and finally to judge eighteen numerical justifications for the same statements. The findings indicated that teachers are well acquainted with numerical examples and counterexamples as proofs. We also found that teachers' considerations for accepting given justifications involve mathematical aspects as well as didactical ones. Teachers are less familiar with students' tendencies to bring more than one example or counterexample in such proofs.