How can I improve my ability to visualize mathematics?

[Mulz]

Hello everyone!

I am having difficulties interpreting mathematical phenomenons visually speaking. I am trying to create an image in my head to explain how the mathematics works, for example, I have tried explaining the term brightness with close results but still flaws.

So how do you become better at "seeing" mathematics? I really want to, do I just draw pictures of it? Where do I even begin in that case? Also, I have severe spatial problems (being very low compared to the average, I can not manipulate objects in my head, only to a certain degree with lots of difficulties), is that maybe why it is harder for me to work on it?

I prefer people that are good at this to explain, or who has been in a similar situation as I am in.

 

[PeroK]

I usually grip the top of my head with both hands, lean forward, put my elbows on the desk, screw my eyes shut and think until my brain hurts!

 

[symbolipoint]

Hello everyone!

I am having difficulties interpreting mathematical phenomenons visually speaking. I am trying to create an image in my head to explain how the mathematics works, for example, I have tried explaining the term brightness with close results but still flaws.

So how do you become better at "seeing" mathematics? I really want to, do I just draw pictures of it? Where do I even begin in that case? Also, I have severe spatial problems (being very low compared to the average, I can not manipulate objects in my head, only to a certain degree with lots of difficulties), is that maybe why it is harder for me to work on it?

I prefer people that are good at this to explain, or who has been in a similar situation as I am in.

Using different wording, the same question has recently been discussed:
https://www.physicsforums.com/posts/4893975/

Being able to manipulate 3-d images all in ones head is an unusual ability, which is why most people construct, form, build models.

 

[homeomorphic]

Practice visualizing whatever you can. It's like lifting weights. You do what you can and progressively give yourself more difficult challenges. You might start out by just visualizing a dot moving around in 2-d, then maybe make it a line. You can get pretty far with just 2-d pictures.

I used to keep trying to visualize things over and over again and eventually I got to the point where I didn't need to practice that much to understand new results, although in certain cases, finding a visual interpretation of some particular things might be a pretty challenging problem that could take a lot of thought. Part of it is carefully choosing exactly what to visualize, so that you don't strain your imagination beyond its limits, so part of the skill is actually learning to work within your limitations and still be able to comprehend things. I was naturally fairly good at visual thinking, but I can say it still took a lot of practice. Studying electromagnetism and reading Visual Complex Analysis and trying to visualize all the arguments in it played a big part. Sometimes, you can even bring in other things besides the visual. For example, in differential geometry, one of my tricks to boost my intuition is to make a certain configuration (a frame of vectors) with my hand and move it around, so I add a kinesthetic sense of how it works. External aids like props, drawings, the diagrams found in textbooks, and computer visualizations can also help. I intend for my next big hobby which I haven't really started on to be using a computer to make mathematical visualizations.

I'm also an artist and used to fold a lot of origami, and it's conceivable that there was some transfer from that. I'm also an amateur pianist (started at age 12), which could have some effect, but I'm only aware of data to this effect for younger children:

http://faculty.washington.edu/demorest/rauscher.pdf

http://www.ncbi.nlm.nih.gov/pubmed/10100200

In general, a thing to watch out for is practicing a certain skill tends not to generalize that much, but there are some striking exceptions to that, such as that study. I'm not sure this means adults should all go and take piano lessons, but it appears to be a good idea for younger children, at least. If you did it with 1st graders, rather than preschoolers, I'm sure there wouldn't be too much difference, but I don't know at what age it will start to be less beneficial.