Understanding Mathematics through Resolution of Paradoxes

Source abstract

Brain-challenging puzzles have attracted people for a very long time. Paradoxes constitute a special type of puzzle aimed to reveal and emphasize an inconsistency or contradiction resulting from some mental experiments in mathematics. Their resolution teaches us to stay alert and be aware of possible flaws of various kinds. Many paradoxes, such as those of Zeno and Russell, greatly influenced the shape of mathematics as we know it today. That suggests a possibility to incorporate the study of paradoxes in standard mathematical courses. But how productive may it be? At which stage of their study will students benefit from being exposed to paradoxes? How one can practically do it in the classroom? This paper is an attempt to address some aspects of these important questions. We discuss the nature and role of paradoxes in the process of understanding, along with potential problems and advantages of their use in study. We give several examples of mathematical paradoxes in both the historical and the classroom context. A short survey results outline an idea of the audience reaction and suggests further directions for research. We conclude that the pedagogical payoff of the use of paradoxes in the classroom is currently underestimated and a comprehensive and consistent study of the impact of paradoxes on learners will allow us to develop a teaching strategy which takes an advantage of the natural curiosity of the mind towards puzzles.