Built-up knowledge evolves over time, because our experiences are always changing. When a new situation or piece of information has to be incorporated into the brain’s intellectual framework, the bricks reshuffle like Tetris pieces. Students need to experience math – not just hear about it, as typically happens in the classroom – to understand it.
Although it seems to be one of the most confounding things in mathematics, infinity can be a gateway drug to deeply personal mathematical experiences. It connects instantly to big, personal questions about life and death, power and control, the beginning of time and the end of the Universe. All of that makes infinity a great focal point for improving Common Core, and for sharpening mathematical literacy in the schools more generally. But really, everyone would benefit from more close encounters of the infinite kind.
In an article for The New York Times, the mathematics teacher Patrick Honner suggested simple ways to bring infinity to kids (or an uncle who thinks he’s bad at math). For the first, all you need is a door. Have the student (or uncle) stand across the room from it. Then, ask them to walk halfway to the door. From their new spot, have them halve the distance between themselves and the door again. And again. And again. How long will it take to reach the door, you ask, if they continue traveling halfway with every step? Even young kids see that the answer to ‘Are we there yet?’ is ‘No, never.’
Demonstrate the same concept by cutting a sheet of paper in half, then in half again, then in half again. The pieces will get smaller – too small to cut. But the paper-cutter can understand that a tiny person with tinier scissors could continue to cut the paper indefinitely – infinitely. Resolving these apparent paradoxes requires the walker or paper-cutter to interpret the novel experience, to build up and shift their mental bricks. This neurological rearrangement represents what Frederic Bartlett, a pioneer of cognitive psychology, dubbed ‘schema’ in his influential 1932 book Remembering: A Study in Experimental and Social Psychology.
Sarah Scoles
https://aeon.co/ideas/how-thinking-about-infinity-changes-kids-brains-on-math
