Pre-Service teachers’ Knowledge of and Beliefs About Direct and Indirect Proofs

Teachers have difficulty integrating proof in their mathematics instruction due to both narrow beliefs about proofs and limited understanding of proofs. Indirect proofs seem to be a particular cause for concern. In this exploratory study, we contribute to the research area by reporting on an empirical study of Norwegian pre-service teachers' knowledge of and beliefs about direct and indirect proofs. Inspired by situativity theory, we investigated pre-service teachers' knowledge of and beliefs about proofs both professed generally and out-of-context and in situation-specific circumstances. Our initial findings are in line with much of the previous literature. First, for situation-specific beliefs and knowledge, we found that indirect proofs seem to be more challenging than direct proofs. Second, for general beliefs and knowledge, we found pre-service teachers' views about proofs in general are narrow and rigid. However, we also investigated possible patterns between general and situation-specific beliefs and knowledge. We found that participants who empirically validated proofs also professed views that a good mathematical argument is an argument that is simply convincing, and not necessarily rigorous. Second, participants who professed preferences for direct proofs, also struggled with the logical conditions of indirect proofs. Implications are discussed.