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 don't know how to use them in my mental concepts for them. I can use very simple formulae like m1v1 = m2v2 (bad example because I don't 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 that's 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 because 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, that's not even auditory thinking that's 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]

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.

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.

Also after questioning many people I've noticed that a lot of people use word definitions to think about things. That baffles me, that's not even auditory thinking that's word thinking because the sounds themselves don't represent concepts its only the words they form that represent concepts.

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]

although algebra can also be tought of visually. I'm very, very bad a thinking in 3D for some reason...

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.

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.

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 realize 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.

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.

Yep that's 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]

Yep that's 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.

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]

How does one think about abstract algebra visually?

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]

... 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 don't know how to use them in my mental concepts for them. I can use very simple formulae like m1v1 = m2v2 (bad example because I don't 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.

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]

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.

I write really slowly and I can't pay attention and write at the same time. I'm going to 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.

How does one think about abstract algebra visually?

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.

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

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 that's not really the case which is why I want to learn to use other modes of thinking).

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 :)

Yep, I don't actually know how to do this though that's 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 want to 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.

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.

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]

I can use very simple formulae like m1v1 = m2v2 (bad example because I don't find that one easy to think about)

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

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.

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.

Are there any visual thinkers here that have learned to think mathematically and use mathematical representations in your mental concepts?

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]

How does one think about abstract algebra visually?

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

 

[ampakine]

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

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:
v = u + at
v2 = u2 + 2as
s = ut + \frac{1}{2}at^2
I can visualise the first one easily but I haven't managed to visualise the other 2 yet.

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.

Thanks, I'm going to try this myself.

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.

Yeah that's 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, that's 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]

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:
v = u + at
v2 = u2 + 2as
s = ut + \frac{1}{2}at^2
I can visualise the first one easily but I haven't managed to visualise the other 2 yet.

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.

Yeah that's 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.

Can you define what you mean by "thinking mathematically"?

 

[Sankaku]

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.

Well said.

 

[ahsanxr]

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

 

[Sankaku]

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

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 .

 

[micromass]

You cannot think visually in mathematics.

If you don't think visually in mathematics, then I'm afraid you won't be able to do much of mathematics. Visual thinking is quite essential in doing mathematics.

 

[jambaugh]

If you don't think visually in mathematics, then I'm afraid you won't be able to do much of mathematics. Visual thinking is quite essential in doing mathematics.

I think your statement is a bit strong. There are two modes of mathematics, visual (geometric) and algebraic. Visualization is excellent for quick understanding but there are many situations where small dimensional coincidences will lead one to err in making higher dimensional generalizations when using visualized analogs or models. At some point one must "do the figures" so to speak. E.g with a rank 6 Lie algebra few (if any) have the skill to visualize the 6 dimensional root diagrams. But you can sit down and churn through their algebraic representation. Good habits and experience with formal language can carry you far without ever invoking any visualization. But the two skills complement.

What I suggest in terms of skills is:
1. By all means practice one's visualization skills. Doodle Euclidean solids, Try to visualize rotations of objects. One might even practice drafting, getting various rotated views using compass, straight edge, T-square and protractor. Finally try to visualize moving around within S3 (the 3-sphere in 4 dimensions). Try visualizing the regular polytopes in 4 dimensions as regular networks in a curved 3 space.

2. Work on using logic rigorously. Study boolean algebra and symbolic logic, using Karnaugh maps to reduce boolean expressions, negating qualifiers, etc. One needs to clearly understand how "A implies B" = "B or Not A", and "Not(A or B) = (Not A) and (Not B)" for examples.

3. Practice carrying out lengthy algebraic calculations in presentation format. I.e. someone should be able to read your sequence of steps and find exactly where you made an error or confirm that you made none. Especially someone should be able to read your work and learn your method.

4. Study combinatorics and linear algebra. They come up so often as to be essential. I would suggest most of the #3 practice be carried out in linear and abstract algebra. Good algebraic habits require you not make e.g. commutativity assumptions (or even associativity assumptions) about algebraic expressions. Learn to expand products and then consciously apply properties to reduce.

Over time one begins thinking in the abstract algebraic language of mathematics while applying visualizations to archtype or analog cases. One's visual skills can guide you heuristically while the formal language gives you the rigor to chart your path through problems.

And finally 5. Breadth of experience gives you interconnections between seemingly different disciplines. One finds that solving linear differential equations is "just" an exercise in linear algebra. One finds similarly much of the "mechanics" of quantum mechanics is just abstract algebra. A little category theory is handy as it is especially aimed at such convergences of disciplines. Play!

Student's now have the entire WWW. I used to walk the stacks at Ga Tech's library picking out books with interesting titles. Even when I couldn't follow 80% of the math, I knew there was a discipline there which related to something which related to something that I did understand. I could then know to go back later when I had more math tools to tackle it. Have fun and apply a bit of audacity. BTW as nice as the web is it still doesn't beat walking the stacks of a library and reading physical books. Quality complete texts aren't broadly available online without shelling out some cash. But now you can sit in the library with a text and look up definitions and notation on wikipedia. That's something I wish I had in my youth.

 

[zahero_2007]

. Quality complete texts aren't broadly available online without shelling out some cash. But now you can sit in the library with a text and look up definitions and notation on wikipedia. .

Actually , nowadays all physics and math textbooks are available on the internet for free .

 

[jambaugh]

Actually , nowadays all physics and math textbooks are available on the internet for free .

Core math and physics textbooks maybe... quality ?, But by "texts" I didn't just mean textbooks. I mean "texts = books" and what I had in mind were books like...
Conway's "On Numbers and Games" or
Mandelbrot's "Fractals"

Two wonderful books I found on the library shelf and thereby was introduce to the subject before I would have run across it in a course or popular article.

By "quality texts" I also mean e.g. a good reference book on Lie Algebras which explains e.g. how to use Young diagrams to enumerate irreps, or say Royden's "Real Analysis" or equivalent. Something like Porteus' "Clifford Algebras and the Classical Groups" which is the only text I've found which clearly explains and enumerates the (correct) spin groups. These are things I've found in the library (and or had to purchase) which you won't find online, although you can find the component topics.

Oh yea, most especially, the Springer series of lecture notes is a fantastic reference but you'll not be getting it online for free.

There's a great deal out there on the web and it grows every day... but there's a bit of crap out there too and it's hard to distinguish crap from diamonds when in electronic format if you're a student. To be sure if you're in a university library, they're not going to buy a great deal of crappy nonsense... they buy what their faculty request.

Besides, you can't beat the bandwidth of an actual well stocked bookshelf! ;)

 

[zahero_2007]

Core math and physics textbooks maybe... quality ?, But by "texts" I didn't just mean textbooks. I mean "texts = books" and what I had in mind were books like...
Conway's "On Numbers and Games" or
Mandelbrot's "Fractals"

Two wonderful books I found on the library shelf and thereby was introduce to the subject before I would have run across it in a course or popular article.

By "quality texts" I also mean e.g. a good reference book on Lie Algebras which explains e.g. how to use Young diagrams to enumerate irreps, or say Royden's "Real Analysis" or equivalent. Something like Porteus' "Clifford Algebras and the Classical Groups" which is the only text I've found which clearly explains and enumerates the (correct) spin groups. These are things I've found in the library (and or had to purchase) which you won't find online, although you can find the component topics.

Oh yea, most especially, the Springer series of lecture notes is a fantastic reference but you'll not be getting it online for free.

There's a great deal out there on the web and it grows every day... but there's a bit of crap out there too and it's hard to distinguish crap from diamonds when in electronic format if you're a student. To be sure if you're in a university library, they're not going to buy a great deal of crappy nonsense... they buy what their faculty request.

Besides, you can't beat the bandwidth of an actual well stocked bookshelf! ;)

Well , All the textbooks you talk about can be downloaded for free including all springer series . One can download all textbooks recommended by university professors from the internet all for free .

 

[ahsanxr]

Actually , nowadays all physics and math textbooks are available on the internet for free .

Certainly not all.

 

[Klockan3]

Certainly not all.

If you don't have a problem with legal issues he is mostly correct, unless your prof took some strange book you will find it.

 

[clope023]

Certainly not all.

I've found Rudin, Apostol, Spivak, Iorodov, Griffiths, Jackson, Goldstein, A'nold, Feynmann, etc among a myriad of literally several thousand (I'm not exaggerating) other physics and math books all in one torrent. It's a pretty close approximation to all.

 

[ahsanxr]

If you don't have a problem with legal issues he is mostly correct, unless your prof took some strange book you will find it.

I'm probably one of the biggest pirates when it comes to movies/music, but I have trouble finding books. For example this semester I wasn't able to find "C Programming" by Brooks, nor "Physics for Scientists and Engineers" by Tipler/Mosca. Is there any specific site which I am missing (if we are allowed to discuss)? Also what mode is it? File sharing sites or torrents?

I've found Rudin, Apostol, Spivak, Iorodov, Griffiths, Jackson, Goldstein, A'nold, Feynmann, etc among a myriad of literally several thousand (I'm not exaggerating) physics and math books all in one torrent. It's a pretty close approximation to all.

Those writers have really well-known books, so I would imagine its easy to find. Textbooks are hard to find when they have newer editions coming out so often.

 

[Geezer]

I've found Rudin, Apostol, Spivak, Iorodov, Griffiths, Jackson, Goldstein, A'nold, Feynmann, etc among a myriad of literally several thousand (I'm not exaggerating) other physics and math books all in one torrent. It's a pretty close approximation to all.

I did, too. Wasn't too hard to find. I downloaded a few books that I didn't already have.

 

[Sankaku]

I don't think this is an appropriate place to be discussing torrent downloads. There was a good discussion here about visual thinking - let's stick to that...

 

[jambaugh]

I've found Rudin, Apostol, Spivak, Iorodov, Griffiths, Jackson, Goldstein, A'nold, Feynmann, etc among a myriad of literally several thousand (I'm not exaggerating) other physics and math books all in one torrent. It's a pretty close approximation to all.

Nearly all of these works are copyrighted. You can also shoplift them from the bookstore or sneak them out of a library. But you cannot LEGALLY download them for free. You may believe this sort of thing does no real damage but it does irreparable harm to one's character to commit the act.

 

[ahsanxr]

Nearly all of these works are copyrighted. You can also shoplift them from the bookstore or sneak them out of a library. But you cannot LEGALLY download them for free. You may believe this sort of thing does no real damage but it does irreparable harm to one's character to commit the act.

Whats the difference between that and borrowing the book from the library? The writer isn't making any money either way.

 

[jambaugh]

Whats the difference between that and borrowing the book from the library? The writer isn't making any money either way.

Character is who you are when no one is looking... and if you don't know why character is important I suggest you think long and hard on the subject.

 

[ahsanxr]

Character is who you are when no one is looking... and if you don't know why character is important I suggest you think long and hard on the subject.

The average cost of a textbook in the US is what an average person makes in a month back home. Poor students like me have to find alternative ways. I don't see how it has to do anything with character. There are those of us who go to the university bookstore and spend 5-600 dollars on brand new books on the first day of classes (my roommate), and then there are people like me who do what they can and buy used books, rent books, get them from the library and download books if the other options are not affordable/available.