Prospective Teachers' Understanding of Proof: What If the Truth Set of an Open Sentence Is Broader than That Covered by the Proof?

This paper investigates prospective elementary and secondary school teachers' understanding of proof in a case where the truth set of an open sentence is broader than the set covered by a valid proof by mathematical induction. This case breaks the boundaries of students' usual experience with proving tasks. The most important finding is that a significant number of students from both groups who recognized correctly the validity of the purported proof thought that it was not possible for the truth set of the open sentence to include any number outside its domain of discourse covered by the proof. The discussion of student difficulties provides insights into the development of instructional practices in teacher preparation programs aiming to uncover these aspects of students' knowledge fragility and address them accordingly. (Contains 3 tables and 1 figure.) [For complete proceedings, see ED496851.]