Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims

This study investigates teachers’ argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible mistakes and explain how they would refute the student's invalid claims. Two of them were also interviewed. The data were analysed in terms of the content and structure of argumentation and the types of counterexamples the teachers generated. The findings show that teachers used two main approaches to refute students’ invalid claims, the use of theory and the use of counterexamples. The role of these approaches in the argumentation process was analysed by Toulmin's model and three types of reasoning emerged that indicate the structure of argumentation in the case of refutation. Concerning the counterexamples, the study shows that few teachers use them in their argumentation and in general they underestimate their value as a proof method.

Research highlights▶ Teachers used mainly theory for refuting students’ invalid claims. ▶ A belief seemed to exist among teachers that refutations by using theorems provide stronger and more general conclusions than by using counterexamples. ▶ The counterexamples provided by the teachers were transparent or specific. ▶ Two cases of reasoning emerged in the case of using only theory: the first one is mathematically wrong and a qualifier is needed while the second one is mathematically correct but pedagogically inappropriate. ▶ Two cases of reasoning emerged in the case of using counterexamples: in the first case only a counterexample was used while in the second both counterexamples and theory.