Math teachers suck

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My brain will treat it all like garbage if it can't make any sense of it. My brain likes to learn, not memorize.

Jamma Jamma is offline

Posts: 366

Re: How does a mathematican picture what he's working on?
This is something that I have always felt- even if I'm dealing with a non-geometric problem, I always have "pictures" in my head of what's going on. These pictures are extremely rough, but I think the brain has a way of trying to "geometerise" everything. It's almost impossible to describe this process though, and I will always do so in different ways, even for the same problem but, for example, if I'm thinking about a quotient group, I may visualise an abstract group as some "blob" partitioned into other "blobs" which interact with each other in a "group-like" way.

That probably makes me sound a little mental, but that's because I'm trying to describe a process that's practically impossible to describe.

Obis Obis is offline

Posts: 17

Re: How does a mathematican picture what he's working on?
I create mental images for the most important, fundamental concepts. For example, I visualize a bijective function similarly to this :

(.................)

||||||||||||

(.................)


Where the top (.................) is some arbitrary domain, the | shows the mapping and the bottom (.................) is the range of the function. I find it very convenient to visualize bijective functions from R^k to R, for example, similarly to this :

(.................)
(.................)
(.................)
(.................)
(.................)

|||||||||||||

(.................)

In this case, an element from the domain is a broken line, that goes through the all the top
(.................).

This mental image is modified at the |||||||||| part if the function is not bijective, for example.

Similarly, I try to create mental images for various fundamental processes and actions in mathematics. Let's say a permutation of 1, 2, 3. The mental image of a permutation of these three symbols is simply a feeling of shuffling these three symbols in some arbitrary way.

Then, visualizing mathematics is an interplay of visualizing the concepts, applying various actions to them while feeling those actions and seeing what is happening.

Old Y, 12:07 AM #4
homeomorphic homeomorphic is offline

Posts: 278

Re: How does a mathematican picture what he's working on?
Being a visual-thinking fanatic, I guess I should give an answer.

How do I visualize it?

I don't know. I just do.

Pretty lame answer, but that's about it. It's hard to say what it's like. It's a lot like what I would draw on a chalkboard. Just figures, often with moving parts. Maybe also feeling pushes and pulls. Sometimes, a sequence might seem like a little animal hopping away.

It often takes a lot of time for me to be able to see the pictures clearly. I sometimes have to keep practicing it, until it becomes easy to see. I think I usually have to practice less these days after so much experience with it. The end result of the process is that many initially non-obvious results become obvious.

Old Y, 01:34 AM #5
chiro chiro is offline

Posts: 1,398

Re: How does a mathematican picture what he's working on?
I agree with homeomorphic.

To me math provides us with a gateway to another kind of sensory perception. With our eyes we are able to represent things more or less in three spatial dimensions, along with a time dimension when you consider spatiotemporal changes.

With mathematics we have a language and a set of tools to allow us to make sense of four, eight, twelve, or even infinitely many dimensions. The tools are still being built, but the ability to have a sensory perception of these kinds of objects is getting a lot more intuitive as we build up results.

After a while you start to think in terms of the language, representation, and usefulness of mathematics because it makes many forms of abstraction found in many other kinds of senses a lot easier.

The best way to get from the non-mathematical to the highly abstract mathematical form of understanding is to use a sensory bridge. Usually we do this by relating a mathematical object, structure, and so on to something that is physical. The physical world is fairly easy for most people to comprehend because we are immersed in it every hour, every day, of every week.

After a while though, the math does have a habit of becoming intuitive which means that at some point you can stop using physical analogs and instead use mathematical analogs, even if they are relatively simple ones.