for the learning of mathematics

an international journal of mathematics education

Boris Koichu - Vol. 32 Num. 1 (2012)
 Enhancing an intellectual need for defining and proving: a case of impossible objects


Identifying mathematical and didactical conditions under which mathematics learners can encounter an intellectual need for defining and proving is recognized as a challenging research enterprise. This paper presents a particular configuration of conditions under which a group of pre-service mathematics teachers successfully constructed a definition of an eminent artistic object, the Penrose tribar, and a proof of its impossibility in 3-D space. The paper focuses on the instructor’s pedagogical choices and interventions, which made the exploration feasible for the students and, at the same time, preserved their autonomous learning. The interventions in the form of auxiliary problems are put forward.