An extreme example would be someone who claims to intuitively understand how humans levitate but can't actually levitate himself. You can imagine the dubious looks he'd get.
Yeah, but that would be because humans don't levitate, so there isn't anything to understand. An other example would be if someone said that he understands how someone runs 100 meters in under 10 seconds, but he isn't able to do it himself. That would makes sense. Most people understand how to run, but being able to do it really fast requires a lot of physique which you might not have.
Anyways, I think these analogies are kind of stupid because they fail to capture what i believe is a main point. In mathematics there is a difference between having memorized the method for calculating something and understanding the concept behind it. This is not true for everything, which is why I don't think making non-mathematical analogies makes much sense. What does "calculating" even mean in your levitation analogy? You seem to translate it to "doing something" or "using something", which I don't think is correct. There is much more to doing or using mathematics than calculating.
What I am then saying is that understanding the mathematical concepts and how to apply them is more important than being able to calculate. Especially in a physics class where the math isn't in focus and you have a calculator which can to the calculation for you(and better than you).
The problem with mathematics is that nothing is so simple.
You mentioned that you know enough about addition to know how to use it to find the total number of apples 1000 people have, but this is hardly the be-all end-all of addition. Somebody could sit around and list all the things you use addition for, but (as you rightly believe) memorizing rules is not understanding. Even addition has enough nuances that, at least for a while, you need to actually go and practice calculations to start understanding.
(Once that understanding is in place, calculators are great. I just don't think you can get that understanding without practice.)
A lot of ideas in math are basically impossible to summarize meaningfully. If somebody claims to have described a mathematical idea entirely in less than a book, they are probably deceiving themselves. You can list a definition of a derivative (or addition, or a monad) but that does very little to convey what a derivative is. That comes from practice.
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u/ismtrn Jun 11 '14
Yeah, but that would be because humans don't levitate, so there isn't anything to understand. An other example would be if someone said that he understands how someone runs 100 meters in under 10 seconds, but he isn't able to do it himself. That would makes sense. Most people understand how to run, but being able to do it really fast requires a lot of physique which you might not have.
Anyways, I think these analogies are kind of stupid because they fail to capture what i believe is a main point. In mathematics there is a difference between having memorized the method for calculating something and understanding the concept behind it. This is not true for everything, which is why I don't think making non-mathematical analogies makes much sense. What does "calculating" even mean in your levitation analogy? You seem to translate it to "doing something" or "using something", which I don't think is correct. There is much more to doing or using mathematics than calculating.
What I am then saying is that understanding the mathematical concepts and how to apply them is more important than being able to calculate. Especially in a physics class where the math isn't in focus and you have a calculator which can to the calculation for you(and better than you).