Revisiting Zeno's Paradoxes: The Evolution of Number Line Models (Pedagogy of Mathematics Series)

By Aileen Velázquez, Omar Hernandez, Jorge Marcial Lopez

Zeno's four paradoxes epitomize the natural tension between discrete and continuous mathematical thinking and succeed in challenging human basic intuitions about space and time. Sorting out the difficulties posed has taken close to twenty-two centuries, and has required the work of the best minds of humankind. Our argument is that Zeno's paradoxes are paradigmatic of the problems that ensue when discrete models are used in lieu of continu- ous ones. We document our presentation with results of tests given to entering students at the Natural Sciences College of University of Puerto Rico at R?o Piedras who evidence similar hesitations as those associated with Zeno's paradoxes, and evidence a lack of understanding of the basic number models of mathematics. Exemplary among these models are: the discrete number lines used to represent natural numbers and integers in arithmetic, the infinitely divisible number lines (Archimedean ordered fields) and the continuum number line (complete or- dered fields). We present a reflection on the evolution of these number line models showing how they build upon each other and propose a unified didactical phenomenology for their mathematical structures, linking them to the mental actions involved in the teaching of arithmetic and measurement. Our approach, while based on an idea of Freudenthal, differs significantly from his approach, and is novel to the extent that it reveals simultaneously, the geometry as well as the relevant arithmetic and algebra. Examples of didactic activities for the classroom are presented to illustrate the teaching of the proposed didactical models.

• Format: Paperback
• Language: English
• ISBN: 1534990224
• ISBN13: 9781534990227
• Release Date: July 2016
• Publisher: CreateSpace Independent Publishing Platform
• Length: 86 Pages
• Weight: 0.62 lbs.
• Dimensions: 11.0" x 0.2" x 8.5"