3 ms·

That it's not at all obvious that a circle maximizes the ratio area/boundary.

sfink 5y ago
It's fairly obvious: first, it's pretty clear that you need to connect all of the fences together to form a single continuous fence (no branches). Second, if you think of it physically, you can take any configuration and pour more "space" (or animals or whatever it's containing) into the enclosed area so that they apply pressure. It's pretty clear that they'll all push outward, and thus that they'll tend towards a circular configuration as you increase the area enclosed. But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof. Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking. There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.
- Tainnor 5y ago
  You're getting downvoted because the same thing happens every time someone wants to fix mathematics education: People who learned it "the old way" insist that there can be no other way. Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).
rasz 5y ago
yes, with all those rectangle balloons etc
- alexanderchr 5y ago
  I agree that the balloon example makes the result believable but the leap from there to “obvious” is enormous.
  - rasz 5y ago
    Isnt if your physics curriculum is aligned.

It's fairly obvious: first, it's pretty clear that you need to connect all of the fences together to form a single continuous fence (no branches). Second, if you think of it physically, you can take any configuration and pour more "space" (or animals or whatever it's containing) into the enclosed area so that they apply pressure. It's pretty clear that they'll all push outward, and thus that they'll tend towards a circular configuration as you increase the area enclosed. But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof. Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking. There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.

Tainnor 5y ago
You're getting downvoted because the same thing happens every time someone wants to fix mathematics education: People who learned it "the old way" insist that there can be no other way. Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).

rasz 5y ago

yes, with all those rectangle balloons etc

alexanderchr 5y ago
I agree that the balloon example makes the result believable but the leap from there to “obvious” is enormous.
- rasz 5y ago
  Isnt if your physics curriculum is aligned.