Sculpting with latex

[reverse]

This is not a calculus question, but rather a question about calculus.

 

I was watching some very bright kids the other day really enjoying calculus.

I have to admit playing with symbols and numbers is not my favourite pastime.

 

When you look at a calculus equation what do you see?

 

Do you see a series of approximations of the real world.

 

Like when I see the sentance “the cat played with the ball ”, I see a picture of a cat playing with a ball.

I do not see the letters or the words.

 

I was wondering ,when you read a calculus equation do you picture,

spheres being intersected by planes?

or

lines folding out to become planes and then extruding to become blocks?

 

Or is it more like solving a crossword puzzle using learned methods?

[Dapthar]

When you look at a calculus equation what do you see?

This is going to be rather unenlightening, but unless the equation is of some surface or function that I'm familiar with, I just see the equation, same as you.

 

Do you see a series of approximations of the real world.

I don't. It would be cool if people who studied Mathematics had "Math-O-Vision", a la Numb3rs, though.

 

I was wondering ,when you read a calculus equation do you picture spheres being intersected by planes and lines folding out to becomes planes and then extruding to become blocks.

Nope. Not me anyways.

 

Or is it more like solving a crossword puzzle using learned methods?

Yup. When I see a problem (in the ideal case), pattern recognition kicks in, and the relevant techniques to solve the problem are recalled from the recalled from the dark recesses of my mind.

 

Of course, when one is working out a proof, this process takes a lot longer than a few moments. For example, a single proof of what I consider to be average difficulty will take me about 15 - 45 minutes of straight thinking, nothing else mind you, just sitting there, trying to get the idea for the proof. However, this is the single most difficult part.

[ydoaPs]

i see simple stuff. like fundamental theorem. [math]{{\int_a}^b}f(x)dx={\lim_{k\to\infty}}{{\sum_{j=1}^k}f(x_j)\Delta{x}[/math]. i see the function over the interval and the area(if it is a familiar function). sorry if the [math]\LaTeX[/math] isn't right, i haven't had much practice with it regarding calc, because i am just starting.

[reverse]

Ok what about the inverse square law, that’s a really good one.

 

explains why you get less spray paint per unit area if you move the can away from a surface.

 

it can be seen as a cone with a point source at the nozzle.

 

I'm sure the inverse square law is really radial.

 

but under the constraint of the circular hole in the nozzle it becomes directional.

 

a cone.

 

so the formula for a cone should be related to the inverse square law if you catch my meaning.

 

so the cone and the inverse square law are cousins if you catch my drift.

 

These are the sorts of Math-o-rama pictures and real world understandings of abstract formulae that I wonder if some people see.

[scguy]

Hmm, i think a fundamental problem with mathematics or should i say the teaching of it is that we are taught how to use systems and procedure to find what we are looking for. If we could actually visualise or understand fully what we are doing then it could be much more fulfilling and easier too. When i am learning things from text books etc it is often the case that instead of explaining the theory and concept properly and extensively it will just rush into the methods without much consideration for the former.

I often stop to just go through the concepts in my head when i am working on a new subject because it gives me insight into how i can apply it better etc. Granted there are gifted individuals that can instantly understand on a very deep level but for others it has to be taught.

[Nicoco]

I don't think it is really useful trying to picture something in the real world when reading mathematics. I mean, 2- and 3-dimensional cases aren't that important most of the time, and I'm not quite able to picture things in n-dimensional space for n>3 (and I don't think most people can)...