My most basic math misunderstanding in school went something like this. Take an equation like “x+1=3”. You can subtract 1 from by both sides and get “x=2″. But why can you subtract 1 from both sides? Like, if two things are “equal”, why can you subtract 1 from both of them and the results will still be “equal”? What does it mean for different things to be “equal” and why does it respect certain operations?
Luckily, my school had math teachers with a clue. They didn’t dismiss my question or answer it outright. Instead they agreed it was a meaningful question that I should solve myself. Eventually I figured it out: expressions are “equal” if they point to the same platonic thing (say, a number within the set of real numbers). When you subtract 1 from both sides of an equation, you’re subtracting 1 from the same platonic thing, so you end up with another platonic thing with two pointers pointing at it.
Shortly after that, emboldened, I approached my physics teacher with another basic confusion: are physical objects open or closed sets? If they are closed (include their border points), how do they touch? And if they are open (don’t include their border points), how do they touch? My physics teacher was also pretty good and replied that we don’t know, but it’s not very relevant to touching, because what we call touching is actually interaction at a distance.
I’ve had many such confusions since, while learning all kinds of things. These confusions seem hard to predict and are rarely emphasized in textbooks, probably because they are personal: everyone has different sticking points and different magic words for overcoming them. That’s part of the reason why explaining stuff to a specific person who can ask questions back feels more productive than writing for a broad audience.
My most basic math misunderstanding in school went something like this. Take an equation like “x+1=3”. You can subtract 1 from by both sides and get “x=2″. But why can you subtract 1 from both sides? Like, if two things are “equal”, why can you subtract 1 from both of them and the results will still be “equal”? What does it mean for different things to be “equal” and why does it respect certain operations?
Luckily, my school had math teachers with a clue. They didn’t dismiss my question or answer it outright. Instead they agreed it was a meaningful question that I should solve myself. Eventually I figured it out: expressions are “equal” if they point to the same platonic thing (say, a number within the set of real numbers). When you subtract 1 from both sides of an equation, you’re subtracting 1 from the same platonic thing, so you end up with another platonic thing with two pointers pointing at it.
Shortly after that, emboldened, I approached my physics teacher with another basic confusion: are physical objects open or closed sets? If they are closed (include their border points), how do they touch? And if they are open (don’t include their border points), how do they touch? My physics teacher was also pretty good and replied that we don’t know, but it’s not very relevant to touching, because what we call touching is actually interaction at a distance.
I’ve had many such confusions since, while learning all kinds of things. These confusions seem hard to predict and are rarely emphasized in textbooks, probably because they are personal: everyone has different sticking points and different magic words for overcoming them. That’s part of the reason why explaining stuff to a specific person who can ask questions back feels more productive than writing for a broad audience.