Difficulties in know ledge integration: Revisiting Zeno's paradox with irrational numbers

The study investigates sources of difficulties exhibited by student teachers in tasks involving the construction of an irrational length segment, and other irrational number tasks. The results show that student teachers know the definitions and characteristics of irrational numbers, yet fail in tasks that require a flexible use of their knowledge and in tasks that involve making connections between different representations. Thus, for example, students say that irrational numbers are real numbers, yet many think they have noplace on the real number line. Their explanations indicate that misconceptions about the limit concept, that relate to the dilemma in one of Zeno's paradoxes, are a main source of difficulty. These findings stress the importance of creating tasks that facilitate the integration of different knowledge pieces.