I came across a set of six "60-second Adventures in Thought" videos by the Open University on thought experiments in philosophy, two of which I think would work well in a math class.
This first mathy video is on Achilles and Tortoise, one of Zeno's Paradoxes. Great for Calculus.
This second video is on Hilbert's Infinite Hotel about infinite sets. At the high school level, it could be a good fit in a math club.
There are more videos linked at Brain Pickings on Grandfather Paradox about time travel, on John Searle's Chinese Room about artificial intelligence, on Twin Paradox about relativity, and on Schrödinger's Cat about quantum mechanics.
(via Vitor Pamplona)