Achilles Overtakes the Turtle: Experiments and Theory Addressing Students' Difficulties with Infinite Processes

The difficulties students have in blending mathematics and physics are here analyzed, by focusing on the issue of a convergent series. We present an experimental and a theoretical analysis of some phenomena which can be investigated employing series, as the bouncing marble and Zeno's paradox of Achilles and the turtle. Measurements were carried out by students employing ICT instruments, such as the smartphone microphone, the smartphone camera or online motion sensors and results were the grounds for a deep discussion about the apparent paradox and the sources of students' misunderstanding. The activities were designed for students on introductory university courses or in advanced high-school classes and was implemented with 90 students of mathematics and physics who are interested in a curriculum addressed to the teaching of mathematics and physics at high school level. Results about their preconceptions before the sequence and some quotes of their metacognitive thinking after the activities are reported.