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The symmetry that makes solving math equations easy

quantamagazine.org

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Re: The symmetry that makes solving math equations easy

#11

Post by anonymouskimmer »

moffkalast wrote: post #9
nightfly wrote: post #4

Earlier quoted context omitted.

Cheating has never need hard to do. Before the internet there were friends/people that could be bribed

Still that's kind of like the "you won't always have a calculator with you" argument. Going to other humans takes time, persuasion and can be expensive. This is a glorified calculator that's free and one click away. Is it really cheating when it's just using another tool in the box? People should learn to do more with everything at their disposal, not arbitrarily limit themselves. Should I not use a 3D printer becaus…

We teach kids in the hopes they will be more able to verify truth or falsity when they can't do the hard work themselves.

This is why black boxes are deprecated in schooling, until the point comes when we believe the students understand enough to be able to verify truth or falsity, at which point we let them use black boxes so that they can learn more. But still, when they are using those black boxes to learn more, we're keeping them from using a black box of the "more". Again, in the hopes that they'll understand the "more" enough to be able to verify the truth or falsity of the results of that "more". Once they can do this, they've hopefully graduated, and are free to use a black box of the "more".

Use black boxes from the get-go and no one realizes that soylent green is people. Or that the Morlocks feed on the Eloi.

Re: The symmetry that makes solving math equations easy

#12

Post by moffkalast »

anonymouskimmer wrote: post #11
moffkalast wrote: post #9

Earlier quoted context omitted.

Still that's kind of like the "you won't always have a calculator with you" argument. Going to other humans takes time, persuasion and can be expensive. This is a glorified calculator that's free and one click away. Is it really cheating when it's just using another tool in the box? People should learn to do more with everything at their disposal, not arbitrarily limit themselves. Should I not use a 3D printer becaus…

We teach kids in the hopes they will be more able to verify truth or falsity when they can't do the hard work themselves. This is why black boxes are deprecated in schooling, until the point comes when we believe the students understand enough to be able to verify truth or falsity, at which point we let them use black boxes so that they can learn more. But still, when they are using those black boxes to learn more, w…

Still if the end goal is to be able to verify the result, learning the entire process is usually not required. Almost everything we use is a black box and while it's good to know the basics to have a good understanding, it's also not a bad idea to strike a balance when dedicating limited learning time between required background and things that are practically useful.

Re: The symmetry that makes solving math equations easy

#13

Post by GuB-42 »

moffkalast wrote: post #9
nightfly wrote: post #4

Earlier quoted context omitted.

Cheating has never need hard to do. Before the internet there were friends/people that could be bribed

Still that's kind of like the "you won't always have a calculator with you" argument. Going to other humans takes time, persuasion and can be expensive. This is a glorified calculator that's free and one click away. Is it really cheating when it's just using another tool in the box? People should learn to do more with everything at their disposal, not arbitrarily limit themselves. Should I not use a 3D printer becaus…

Solving exercises have never been a practical problem. The solution is already known by the one asking the question, the only point is training and skill assessment.

The point of solving these exercises is for you to train yourself in manipulating functions, maybe so that you can later do it in more complex and novel situations where neither ChatGPT nor WolframAlpha will find the solution for you (or worse, give you a wrong answer). It doesn't mean they are useless tools, actual scientists and engineers use them, but because they actually solved the exercises when they learned to do maths, they are able to recognize when they fail and adjust their queries to have they do what they need.

And while the "you won't always have a calculator with you" is mostly wrong nowadays. And it also was before smartphones since pocket calculators were extremely cheap and small so you could always have one on you if you wanted to, and they were actually more reliable than smartphones. What is important is not to put your entire trust into the calculator. You have to recognize that if you calculator tells you that $42 + 10% is $63, you messed up somewhere. It is the same with ChatGPT, it can help you, but if you don't know your subject, you won't be able to differentiate between helpful answers and hallucinations, making it worse than useless.

Re: The symmetry that makes solving math equations easy

#15

Post by chongli »

onos wrote: post #6
svantana wrote: post #5

I feel like this article is a bit backwards - solving quadratic equations is only easy if you have access to the square root function, which by definition is a solver of quadratic expressions. Without it, one needs to resort to iterative root-finding, which works for polynomials of any order.

That’s not right. For example, if you have access to any sort of radical you still can’t solve the quintic.

Just use Newton's method!

Re: The symmetry that makes solving math equations easy

#16

Post by archgoon »

gweinberg wrote: post #14

It's fortunate that we so seldom need to solve cubic or higher order polynomial equations. Quadratic equations come up in physics all the time, because it's very common to have a constant acceleration. But when does one ever come across a constant jerk?

[dead]

Re: The symmetry that makes solving math equations easy

#17

Post by justeleblanc »

chongli wrote: post #15
onos wrote: post #6

Earlier quoted context omitted.

That’s not right. For example, if you have access to any sort of radical you still can’t solve the quintic.

Just use Newton's method!

I can't tell if this is a joke. Newton's method will give you an approximate answer.

Re: The symmetry that makes solving math equations easy

#18

Post by drc500free »

gweinberg wrote: post #14

It's fortunate that we so seldom need to solve cubic or higher order polynomial equations. Quadratic equations come up in physics all the time, because it's very common to have a constant acceleration. But when does one ever come across a constant jerk?

I've worked with several.

Re: The symmetry that makes solving math equations easy

#19

Post by gweinberg »

nightfly wrote: post #4
photochemsyn wrote: post #2

Nice write-up! Incidentally I ran all the exercises through ChatGPT which seems to have completely crushed them, with work shown. Forget about assigning grades via math homework, it only took a few minutes (though I should go through and check it by hand I think). [edit looking at the answers in more detail it seems to go down some strange rabbit holes that probably were not what the author intended] Also, if one wan…

Cheating has never need hard to do. Before the internet there were friends/people that could be bribed

Getting help on homework isn't really cheating. I had one professor who sort of teased the class when we admitted we were mostly working on homework alone. He felt we wers suspposed to be working in groups. I think the only reason homework is graded at all is because if it wasn't, teachers assume the students wouldn't do it. The tests are there to test you. The homework is just there for practice. But if you just let someone else do your homework, you're unlikely to pass the test. Conversely, if you can score well on the tests without doing the homework, you shouldn't have to do it, it's just a waste of time.
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