Training yourself to think mathematically (or visually)

ampakine

In college I struggled greatly with physical chemistry lab reports but found the organic and inorganic lab reports easy. I also struggled through the physical chem labs in general and struggled with the 1st year general chem labs and I've concluded that its all about my thinking style. I am very good at visual thinking and as a result I use it whenever I can and this thinking style is excellent for learning many concepts in chemistry especially inorganic chemistry concepts and a lot of organic chemistry concepts but when it comes to physical chemistry I find myself struggling because in the explanations I'm reading I find mathematical formulas in the place of pictures and I don't know what to do with these formulas, I dont know how to use them in my mental concepts for them. I can use very simple formulae like m1v1 = m2v2 (bad example because I dont find that one easy to think about) in my mental concepts because I can visualise each variable representing a proportion in my mental images and in my head I can watch the mental image change as I change the value of a variable.

My head hurts just thinking about attempting to create a mental representation of formulae like the Schrodinger equation so I've hit a bit of a road block. People do think about these concepts though so whatever they are doing, I should be able to do. I've asked people about this and it seems that the people who find these unthinkable (I mean the ones I struggle with) concepts easy are usually the ones who find visual concepts difficult and after questioning them about how they do it I've concluded that they somehow think mathematically. I'm sure this forum is full of mathematical thinkers but I can't just ask you how you do it because thats like asking someone "how do you think" or "how do you breath" which is a hard question to answer because you just do it. Are there any visual thinkers here that have learned to think mathematically and use mathematical representations in your mental concepts? Conversely are there any mathematical thinkers here who learned to think visually? One of my friends seems to be a predominantly mathematical thinker and he has trouble with all the concepts I find easy and vice versa. I have no idea how to describe to him how to start thinking visually cuz its something I just do without knowing how I do it. Also after questioning many people I've noticed that a lot of people use word definitions to think about things. That baffles me, thats not even auditory thinking thats word thinking because the sounds themselves don't represent concepts its only the words they form that represent concepts.

micromass

I think I'm also quite a visual thinker. When I see something mathematically, I immediately try to visualize what's going on. This probably explains my natural preference to analysis and topology, although algebra can also be tought of visually. I'm very, very bad a thinking in 3D for some reason...

Now, when I was a freshman, we first saw the concept of a quotient group, and I really didn't understand it. At least, I did understand, but I couldn't visualize what's going on. So thought about it and thought about it and I made extremely complicated mental pictures of what's going on. Then I gave it a rest for a week or two, and after that time I understood everything. The point was that I had given up thinking visually about it, since it didn't bring me anywhere. Somehow, making a lot of exercises about it helped me create a new visual imagery that was less intuitive, but somehow formed out of experience.

So, my advice to you is to make a lot of exercises with the formulae until they become second nature. Once you get used to it, you will gain an intuition for it and you will start seeing things visually...

That's my experience as a mathematician, I hope it also can be helpful to physicists...

Geezer

Can you "see" gradients, vector curls, and the like?

With experience, I've learned to "read" equations (e.g., recognize equations describing waves). Taking Classical Mechanics has helped a lot with that.

My brain is FUBAR, so how I deal with information is most likely not how anyone else does it. I'm entirely visual. When people speak to me, my brain has to translate the spoken words--the sounds--into [visual] subtitles and then I "read" the subtitles...that's how I "listen." Seriously. There is a part of my field of vision set aside where these subtitles appear. If there are too many auditory distractions or if the speaker speaks too fast, my brain can't translate the words right, and I end up missing everything that was spoken.

So, as far as equations are concerned, my visual memory is so strong that I can just "see" the equations...I don't really have to put any effort into memorizing them. That said, having the equations at my disposal doesn't necessarily mean I automatically know how to use them. That just comes with practice.

ampakine

Yeah you can visualise the proportions. Whenever I read about visual thinking, every explanation claims that visual thinkers do poorly in algebra but algebra is one of my strongest areas in maths. I'm better at algebra than I am at geometry.

Strangely enough I know exactly what you're talking about. This happens to me all the time but I have no idea why. I struggle and put enormous amounts of effort into coming up with a visual representation of the concept and eventually give up and I assume that I accomplished nothing because I never managed to come up with a complete visual representation but days later I realise I have a profound understanding of the concept that I could only have gotten from all that effort I had put into visualising the concept. Its as if I exercised my brain with all that effort and caused it to form new connections that allow me to think in ways I was previously uncapable of thinking. I'd made this observation before but I hadn't really thought about it until you brought it up there. This seems to be a solid way of increasing your mental capacity. This is how I became good at algebra. I haven't attempted to do it with calculus yet but I bet it would work quite well with that.

Yep thats how it is for me too. I often struggle in lectures because I don't have time to translate the words into images. On top of that I need to visualise various different aspects of concepts in order to solidify them in my mind and that takes time. Simple phrases have become second nature to me so they're automatically translated by my brain but when it comes to unfamiliar scientific concepts I need to consciously translate everything.

Leveret

Have you tried reviewing your notes immediately after lectures? If you can copy down enough of what your professor says to recall the gist of it from your short-term memory, you can go back over it at your own pace and make whatever mental images you need.

Skrew

How does one think about abstract algebra visually?

chiro

In terms of groups, one way is to think about things like say a chess board or a rubix cube. Both of these things allow you to perform operations and also "undo" them just like you can do with group operations and inverses.

So in that sense, systems that have this property do have an analogy to the chessboard or the rubix cube.

mal4mac

Why do you feel the need to form a mental image of everything?

Hawking in "Universe in a Nutshell" says he has difficulty visualising things in three dimensions, never mind 4 (or 10 or 11!) So he uses mathematics as a way to get away from having to visualise!

Try reading "Mathematics:A very short introduction" by Timothy Gowers. It may get you away from this "visualisation" approach, when what you need is an "abstract rule following" approach. I use m1v1 = m2v2 to find m1 following a snappy algebraic rule ("m1 = m2v2/v1") not by trying to visualise the situation ... not sure I can even do it that way!

Visualising space-time as the surface of a sphere is fun and can provide some useful intuitions, but it's (i) incomplete (ii) not easy to calculate with. To do serious calculations you need the equations, which you can't visualise. But you can plug in numbers and get an answer by following the rules ... no visualising required. Fortunately that's all you need! Realise that and physics becomes easy... no mind bending needed... even a chemist can do it :)

Trying to visualise what is going on with the Schrodinger equation is impossible - no one can understand it (as Feynman said.) This is equivalent to saying no one can visualise it.

Plug and chug may seems a bit dry and abstract compared to all that fun visualisation, but that's physics. If you want wet and practical stick to experimental chemistry...

ampakine

I write really slowly and I can't pay attention and write at the same time. I'm gonna start recording the lectures with my phone or something. If I could hook an external harddrive up to my phone then space wouldn't be an issue.

Its not actually abstract, the variables represent proportions. I've done a fair bit of programming so I have a fair bit of intuition about variables and functions though that might be why I can visualise algebra. Learning to program I had to find ways to visualise everything which was pretty hard but when I finally figured it out programming became easy and I got good at it real fast. For example loops were always a mystery to me, I had an idea of what they did but I just couldn't get my head around them but I started putting lots of effort into trying to visualise how they work and eventually it became second nature to me. Now I just see a loop as a big list that expands out of a single line depending on what parameters/variables its fed. Functions I see as a kind of machine, you feed variables/parameters into the left hand side of it and it spits out structurally formatted stuff on the right hand side. Its a very vague, transparent mental image that doesn't overlap with my actual vision but its highly effective at understanding things in programming work.

I don't know. Habit maybe. I want to learn how to think mathematically though because there are many things that cannot be visualised but can be arrived at through mathematics. I'd like to be able to switch between modes of thinking because the way I see it that would vastly expand my mental capacity. Its not like an English speaker learning Russian, its more like someone who only has eyes gaining a pair of ears. Like I said I can only visualise simple equations so its mathematical intuition I'm using for more complicated ones but I can't use that as a mode of thinking about things other than algebra. I can use visualisation to think about just about anything (well thats not really the case which is why I want to learn to use other modes of thinking).

Yep, I don't actually know how to do this though thats why I'm asking questions. To do mental algebra I actually see the equation. When you do it which of the 5 senses (if any) do you use? In the mind we're not limited to the 5 senses, we also have emotion which is definitely not the same thing as touch. Although the majority of my thinking is visual, I attach other stuff to it for example to think about magnetism I attach the feeling of attraction and repulsion which is the tactile sense. I never use sound though, besides thinking about things that make sounds I never really use sound to think about anything. This might all be a matter of habit. I wanna learn to use my right hand to write so I'll hand 2 different hand writing styles and signatures but since I'm so terrible with that hand I have to constantly force myself to keep using it otherwise I'll switch to my good hand.

In the documentary "Atom" they made it out like Einstein, Schrodinger and some other conventional physicists hated the theories of Bohr and Heisenberg because they were entirely non visual and thus incomprehensible to visual thinkers. I'm not surprised at all, I'm glad that squared wavefunctions represent 3D spaces that I can picture in my head because if orbitals were only represented by mathematical equations they would be meaningless to me. This is clearly a disability I have, Bohr and Heisenberg were able to think about these things without images but I wouldn't know where to begin. I'm aware that the true nature of the atom is something that cannot be visualised or thought about with any of the 5 senses but can more than likely be arrived at through mathematics.

Klockan3

How is it hard to picture that the larger an object is the slower it moves?

I like mental pictures as well, for the Schrödinger in 1d I view its solutions as vibrating strings in a 3d space with 2 of the dimensions being a complex plane while the last is the spatial dimension. You can extend this to two and three dimensions to get higher dimensional pseudo surfaces. With that a free particle in 1d becomes a helix which rotates forward in time as an example, while in 2d it becomes a continuum of helixes.
I naturally think in both ways, I try to get as many ways to think about each subject as possible and that includes both getting mental pictures and getting a mathematical sense of how it all works out and them making both of these click together.

Simfish

http://www.central.edu/eaam/

ampakine

Sorry I meant M1V1 = M2V2 where M is the concentration of a solute and V is the volume of the solvent. This isn't actually hard to picture, you just use colour density to gauge the concentration. Either that or visualise the particles growing in number as the volume decreases. A better example though is the equations for motion:
[tex]v = u + at[/tex]
[tex]v2 = u2 + 2as[/tex]
[tex]s = ut + \frac{1}{2}at^2[/tex]
I can visualise the first one easily but I haven't managed to visualise the other 2 yet.

Thanks, I'm gonna try this myself.

Yeah thats what I'm trying to do myself. I do often approach concepts mathematically to get additional info that I wouldn't have gained by visualising but what I'm talking about is using mathematical thinking as a complete mode of thinking that you can use to think about everything not just concepts in physics or chemistry or whatever. I use visual thinking for everything in my daily life, its my main mode of thinking but I'm trying to figure out how to switch to mathematical thinking if I need to. From questioning various people about how they think it seems that there also exists a mode of thinking that involves words. Verbal thinking. People use words and descriptions to think about things. I don't understand how that works either. If you think in words then your thinking could only be as complete as your vocabulary. I suspect that they are really thinking visually but they've had the habit of speaking inside their head for so long that it makes it has become automatic and makes it difficult for them to observe how they think. Although I keep saying I think visually, thats not fully the case. The other senses are involved, particularly tactile sense. Sound and smell only comes into it when I'm thinking about things that make sounds or smell though.

Klockan3

To me the picture for all of those is the same thing, that is the strength of mathematical thinking. Mathematics is about equivalences and thus allows you to merge equivalent mental pictures into stronger ones and allows you to abstract pictures to cover more topics.
Can you define what you mean by "thinking mathematically"?

Sankaku

Well said.

ahsanxr

Not every relation is simple enough to visualize/have an intuition for.

Sankaku

Simple? No.

Possible? Often...

If it was simple, it wouldn't be nearly as rewarding.

:-)

zahero_2007

You cannot think visually in mathematics . you can only think logically because mathematics is nothing but logic . you must train yourself to think in this way for example when I think about schrodinger equation , I do not try to visualize it but I try to see ways so that it can be solved .