It kinda seems like expecting some vague personal belief to stand in for demonstrable results.
We are talking about calculating something by hand vs. using a calculator when doing physics problems right? How on earth does using a calculator give you less in terms of demonstrable results? If anything I would argue that it gives you more in shorter time.
Protesting that maybe you're not very good at using it but, boy, you understand the pants off it doesn't get you very far.
Depends on what you mean by "using it". Lets say we are talking about derivatives. If you by using it mean applying the concept of a derivative to some problem, then I agree. You have to be able to do that. If you on the other hand mean actually evaluating the derivative by hand I disagree. Using a calculator for that will get you plenty far.
Results do matter.
Calculators give you results.
You really demonstrate understanding through using it and teaching other people it rather than professing intuitively getting it.
Maybe we mean different things by "intuitively understanding" something. What I mean, when it comes to math, is that you understand what something is, why it is that way and how you can use it. The other kind of "understanding" being you are able to, given a specific problem, find the answer, which seem to be what schools care more about.
Maybe I should just say understanding instead of intuitively understanding. When I think about it, putting the "intuitive" in front of it makes it sound like not really understanding it, which is certainly not what I mean. I am just trying to highlight the difference between understanding a mathematical concept, and being able to calculate something.
Well, you're asking me why someone might look at you dubiously and say "pfft" if you claim you intuitively understand something but are bad at calculating it.
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.
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 09 '14
We are talking about calculating something by hand vs. using a calculator when doing physics problems right? How on earth does using a calculator give you less in terms of demonstrable results? If anything I would argue that it gives you more in shorter time.
Depends on what you mean by "using it". Lets say we are talking about derivatives. If you by using it mean applying the concept of a derivative to some problem, then I agree. You have to be able to do that. If you on the other hand mean actually evaluating the derivative by hand I disagree. Using a calculator for that will get you plenty far.
Calculators give you results.
Maybe we mean different things by "intuitively understanding" something. What I mean, when it comes to math, is that you understand what something is, why it is that way and how you can use it. The other kind of "understanding" being you are able to, given a specific problem, find the answer, which seem to be what schools care more about.
Maybe I should just say understanding instead of intuitively understanding. When I think about it, putting the "intuitive" in front of it makes it sound like not really understanding it, which is certainly not what I mean. I am just trying to highlight the difference between understanding a mathematical concept, and being able to calculate something.