This dissertation reports on an investigation of transition-to-proof students' understanding of proof by contradiction. A plethora of research on the construction aspect of proof by contradiction is available and suggests that the method is one of the most difficult for students to construct and comprehend. However, there is little research on the students' comprehension of proofs and, in particular, proofs by contradiction. This study aims to fill this gap in the literature. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, this study proposes a preliminary genetic decomposition for how a student might construct the concept `proof by contradiction' and a series of five teaching interventions based on this preliminary genetic decomposition. Data was analyzed in two ways: (1) group analysis of the first two teaching interventions to consider students' initial conceptions of the proof method and (2) case study analysis of two individuals to consider how students' understanding developed over time. The genetic decomposition and teaching interventions were then revised based on the results of the data analysis. This study concludes with implications for teaching the concept of proof by contradiction and suggestions for further research on the topic.

Contro l’argomento anselmiano del Proslogion sono state sollevate varie obiezioni, come quelle, molto note, di Gaunilone, di san Tommaso e di Kant. In questo saggio si sostiene la tesi che tutte queste critiche si basano fondamentalmente su una interpretazione imprecisa del testo di Anselmo che, se correttamente letto, ne risulta invece al riparo. Viene quindi offerta una nuova lettura dell’unum argumentum, con la quale, ricercandone lo spirito originario e più autentico, viene messa in risalto innanzitutto la sua struttura formale di dimostrazione per assurdo, illustrando in secondo luogo l’importante rapporto di complementarità che lega questa tipologia di prova a quelle strutturate in modo diverso, che procedono cioè a posteriori per costruzione diretta. / Many important Authors – as Gaunilo, Thomas Aquinas and Kant – have brought many well-known objections against the anselmian argument. This paper proposes the thesis that all these objections are based on an inaccurate interpretation of the Proslogion: in fact, an in-depth analysis of the text shows its fully validity. Therefore, it is offered a new reading of the anselmian argument, that looks for its original and authentic sense: firstly, it is enlightened its formal structure of proof by contradiction, and secondly it is showed its important complementary relationship with the other kind of a posteriori proofs.