How Can I Improve My Math Skills as a University Freshman?

[wlcgeek]

I'm fascinated by math (uni. freshmen), but really haven't taken any hard math courses and feel kind of slow at it (highest class was high school calculus).

Can someone get "good" at math? Many people say that it's more of an innate talent like sports or music. You can only improve so much and that to be really good you have to have that talent.

What do people think about this argument?

And what are some ways or tips to improve - even if perhaps I might not be able to be great at math (e.g. getting a master's or Ph.D. in it). How does one even go about thinking about math? I feel often slow at it and not sure where to start. In fact, my brain hurts sometimes thinking about math (whereas I don't experience that with studying something like English or biology), but I actually like the results if that makes sense.

It's like solving a puzzle. It can be arduous, but the end result is very enjoyable if you get the right answer. I just wish I were better at it and don't know how to improve (if that's possible).

Appreciate everyone's thoughts!

 

[HallsofIvy]

To learn math, you have to do math- do lots and lots of math problems starting with some basic algebra problems and working up. Be sure to check your answer in the original problems.

 

[homeomorphic]

Can someone get "good" at math?

Yes, no one starts out being "good" at math. The only people who are good got good. Can ANYONE get good? I don't know. To an extent, yes.

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

Many people say that it's more of an innate talent like sports or music. You can only improve so much and that to be really good you have to have that talent.

In sports, it's more clear how genes are involved simply because things like muscles, height, and so on are much simpler than the brain. In intellectual pursuits, it's not very clear how genes are involved because we don't understand the brain very well. Who says music is an innate talent? It's in how you practice. Practice the right way and you improve. Practice the wrong way and you won't improve much and you'll pick up bad habits. I play piano and with that, it really takes a lot of willpower to get yourself to practice the right way. If you just let go and play by instinct, it's no good. There have even been times when I have been embarrassed to play the same way in front of my teacher because I knew I was practicing the wrong way out of laziness.

If you have ever worked through Drawing on the Right Side of the Brain, you'll find that many people can actually draw well if they just look at things the right way. Some people might practice drawing their whole life, and someone else with the same ability could outperform them in a matter of days if they just read Drawing on the Right Side of the Brain. I am not sure that is exactly what would happen. Maybe if you just tried to draw by trial and error, eventually, you'd catch on, but it's conceivable that you might never figure it out. One person knows the trick, the other person doesn't and that makes all the difference.

As far as math goes, I have a very mathematical family, so I do think genetics have something to do with my abilities (there might be some environmental influences, but it's not like my parents ever really sat down and taught me math or talked about it with me). However, there are plenty of tricks I picked up, too, just like Drawing on the Right Side of the Brain, where if you know the tricks of the trade, it is a big part of being good at it. Different people may need different tricks. Some people may need to do different things than I did because maybe some things don't come as naturally to them or maybe other things come more naturally than for me.

 

[jimmyly]

Practice practice PRACTICE. I went from a B in math to an F from grade 8-11 because i stopped doing homework and going to class. Now i have an A+ and just got 100% in my last test in precalculus12. Practice practice and more practice is all i can say.

 

[FreeMitya]

Practice practice PRACTICE. I went from a B in math to an F from grade 8-11 because i stopped doing homework and going to class. Now i have an A+ and just got 100% in my last test in precalculus12. Practice practice and more practice is all i can say.

Brian Cox, at one point, was getting D's in high school maths because of a lack of interest and, in turn, practice.

 

[azizlwl]

Yes, no one starts out being "good" at math. The only people who are good got good. Can ANYONE get good? I don't know. To an extent, yes.

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

In sports, it's more clear how genes are involved simply because things like muscles, height, and so on are much simpler than the brain. In intellectual pursuits, it's not very clear how genes are involved because we don't understand the brain very well. Who says music is an innate talent? It's in how you practice. Practice the right way and you improve. Practice the wrong way and you won't improve much and you'll pick up bad habits. I play piano and with that, it really takes a lot of willpower to get yourself to practice the right way. If you just let go and play by instinct, it's no good. There have even been times when I have been embarrassed to play the same way in front of my teacher because I knew I was practicing the wrong way out of laziness.

If you have ever worked through Drawing on the Right Side of the Brain, you'll find that many people can actually draw well if they just look at things the right way. Some people might practice drawing their whole life, and someone else with the same ability could outperform them in a matter of days if they just read Drawing on the Right Side of the Brain. I am not sure that is exactly what would happen. Maybe if you just tried to draw by trial and error, eventually, you'd catch on, but it's conceivable that you might never figure it out. One person knows the trick, the other person doesn't and that makes all the difference.

As far as math goes, I have a very mathematical family, so I do think genetics have something to do with my abilities (there might be some environmental influences, but it's not like my parents ever really sat down and taught me math or talked about it with me). However, there are plenty of tricks I picked up, too, just like Drawing on the Right Side of the Brain, where if you know the tricks of the trade, it is a big part of being good at it. Different people may need different tricks. Some people may need to do different things than I did because maybe some things don't come as naturally to them or maybe other things come more naturally than for me.

Very interesting.
I am very very slow leaner but i love Physics and Maths and music(not talented, anyway i keep practicing my piano and i will forget the last song when i learn a new one).
I know i will never excel in them but doing the practices makes me happy.

Maybe you can elaborate on "Drawing on the Right Side of the Brain". Any books you wish to suggest?

 

[jimmyly]

Brian Cox, at one point, was getting D's in high school maths because of a lack of interest and, in turn, practice.

i did not know who he was until you mentioned him! that gives me more motivation to pursue my dreams. i was not interested in academics until i fell in love with physics. Now i can't get enough of math and physics and find myself always thinking of different equations and always calculating stuff in my head. i find myself smiling during lectures and while reading my textbooks because of how fascinated i am.

Very interesting.
I am very very slow leaner but i love Physics and Maths and music(not talented, anyway i keep practicing my piano and i will forget the last song when i learn a new one).
I know i will never excel in them but doing the practices makes me happy.

Maybe you can elaborate on "Drawing on the Right Side of the Brain". Any books you wish to suggest?

don't sell yourself short. i think one of the greatest qualities to have is passion. if you are passionate about something and dedicated, you will be amazing. Like everything it takes time(sometimes a LOT of time) to excel.

if something is easy, it is not worth doing. Also, if something is easy, everyone would do it.

i think the reason why i love math and physics IS because it is difficult. i have been told i am talented at everything. i taught myself guitar at 11yrs old, i excel at any sport first try, English and writing is a breeze to me. All those things are extremely easy to me, that is why i left them behind and don't do as much anymore.(i don't mean to come off as conceited)

But Math and Physics are a challenge. They may NEVER be understood by any human. That is the beauty i find in it. NO ONE will ever understand the universe and everything within. All we can do is learn more and more, day after day.

i went a little off topic lol but bottom line

practice, practice, PRACTICE
passion, dedication, and motivation are your bestfriends

 

[member 392791]

It's like anything else in life...you might be born with a slight advantage, but if you don't practice it's meaningless

Kobe Bryant didn't get in the NBA on pure talent and not practicing...he was out there shooting hoops everyday I'm sure. Combined with natural talent is a force to be reckoned with. But if you don't have natural talent, you can atleast train yourself to a certain level, certainly enough math for most professions.

 

[Group_Complex]

I'm fascinated by math (uni. freshmen), but really haven't taken any hard math courses and feel kind of slow at it (highest class was high school calculus).

Can someone get "good" at math? Many people say that it's more of an innate talent like sports or music. You can only improve so much and that to be really good you have to have that talent.

What do people think about this argument?

And what are some ways or tips to improve - even if perhaps I might not be able to be great at math (e.g. getting a master's or Ph.D. in it). How does one even go about thinking about math? I feel often slow at it and not sure where to start. In fact, my brain hurts sometimes thinking about math (whereas I don't experience that with studying something like English or biology), but I actually like the results if that makes sense.

It's like solving a puzzle. It can be arduous, but the end result is very enjoyable if you get the right answer. I just wish I were better at it and don't know how to improve (if that's possible).

Appreciate everyone's thoughts!

It depends how far you want to go. I once thought I would breeze through everything up to my phd and win the fields medal, because I always got 100% in math exams and did quite well on IQ tests etc. Now that I am at university (I admit I took calculus and linear algebra in High School) I have found that there is a large degree of innate talent which is required to stand out ahead of your peers. If you just want to do well in calculus or elementary analysis I would say study and dedication will help you get excellent marks on the finals, but if you want to do something original in mathematics (which is the real meaning of being good at mathematics) then I would say no, you probably can't do it if you are having problems with calculus.

 

[homeomorphic]

Actually, I had math prof who tells the story of how he got interested in math because he FAILED calculus and that motivated him to overcome his difficulties. He went on to be an analyst at what is now a top 20 math dept, according to some rankings.

 

[jimmyly]

It depends how far you want to go. I once thought I would breeze through everything up to my phd and win the fields medal, because I always got 100% in math exams and did quite well on IQ tests etc. Now that I am at university (I admit I took calculus and linear algebra in High School) I have found that there is a large degree of innate talent which is required to stand out ahead of your peers. If you just want to do well in calculus or elementary analysis I would say study and dedication will help you get excellent marks on the finals, but if you want to do something original in mathematics (which is the real meaning of being good at mathematics) then I would say no, you probably can't do it if you are having problems with calculus.

Yes but anyone can improve and get good at something. Just because someone is taking a little bit longer with something doesn't mean they won't do any original work. Its when they get good then they can create their own work. I may not know as much because i admit i messed up and it set me back a couple years but i know everyone learns at a different rate and eventually when one has the solid foundations and mastered it, they will be able to expand further and create their own contributions to science.

 

[SolsticeFire]

NO ONE will ever understand the universe and everything within,

If you really believe that nobody will ever understand the universe then I'm afraid your undertaking is not sincere; you merely are fascinated by the intellectual stimulation physics and math gives you. If your goal is not to understand the universe in its entirety and seek out the true meaning behind things then why bother with technicalities?

If all mathematicians and physicists thought that NO ONE will ever understand the universe and everything within then there wouldn't be such vigorous progress towards the truth, both individually and as mankind. If you aren't prepared to go all the way, then don't go at all.



Ohh and I know this really great quote that summarizes my thoughts on the topic: :D

A taste for truth at any cost is a passion which spares nothing. - Albert Camus

SolsticeFire

 

[jimmyly]

If you really believe that nobody will ever understand the universe then I'm afraid your undertaking is not sincere; you merely are fascinated by the intellectual stimulation physics and math gives you. If your goal is not to understand the universe in its entirety and seek out the true meaning behind things then why bother with technicalities?

If all mathematicians and physicists thought that NO ONE will ever understand the universe and everything within then there wouldn't be such vigorous progress towards the truth, both individually and as mankind. If you aren't prepared to go all the way, then don't go at all.



Ohh and I know this really great quote that summarizes my thoughts on the topic: :D

A taste for truth at any cost is a passion which spares nothing. - Albert Camus

SolsticeFire

if any human thinks they can understand everything that goes on in the universe they're ignorant. we are one of many very insignificant species which our brain cannot process many many things. We know how things work, but that's all. what i mean by understanding the universe is that you have a clear understanding of say, gravity or magnetism. What is that force? not describing what it does, but what IS it? and WHY does these things do what they do. and it is extremely difficult to answer those questions because these things are so bizarre.

don't get me wrong, i WANT to know everything. but here is something to think about. Once someone has discovered EVERYTHING and understand the universe and everything within. That my friend, will be the end of physics. There will be no more reason to go on searching for answers if everything is known.

the fun and beauty of physics IS because it is mysterious. i get shivers down my spine learning about even atoms which is taught to every high school student. My goal is to learn and understand as much as i can.

since you ended your post with a quote, as will i :P lol

"but I don't have to know an answer. I don't feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose which is the way it really is as far as I can tell."
-- Richard P. Feynman

The universe is not MADE for humans to understand.

 

[jimmyly]

if any human thinks they can understand everything that goes on in the universe they're ignorant. we are one of many very insignificant species which our brain cannot process many many things. We know how things work, but that's all. what i mean by understanding the universe is that you have a clear understanding of say, gravity or magnetism. What is that force? not describing what it does, but what IS it? and WHY does these things do what they do. and it is extremely difficult to answer those questions because these things are so bizarre.

don't get me wrong, i WANT to know everything. but here is something to think about. Once someone has discovered EVERYTHING and understand the universe and everything within. That my friend, will be the end of physics. There will be no more reason to go on searching for answers if everything is known.

the fun and beauty of physics IS because it is mysterious. i get shivers down my spine learning about even atoms which is taught to every high school student. My goal is to learn and understand as much as i can.

since you ended your post with a quote, as will i :P lol

"but I don't have to know an answer. I don't feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose which is the way it really is as far as I can tell."
-- Richard P. Feynman

The universe is not MADE for humans to understand.

I just thought of something interesting so to add to my last post.

So suppose you met a beautiful girl. You don't know nothing about her. So you ask questions to find out more. By the end of it, all you have are things you THINK she has told you. But you really don't understand her at all. You don't know what she's thinking, what she's going to do. You don't know why she does the things she does. This girl is mother nature. Something beyond what our human minds are capable of processing. We can speak to her through mathematics. The language she speaks in. But we really don't know a damn thing about her. Only the rules that she lives by.

You may mean understanding the universe by the laws which govern them. If that's the case, maybe just maybe one day humans will find out all the rules. But i do not think so because of the fact that science keeps discovering more and more. I was talking to a friend who is extremely bright and has just complete his phd. He said the more we learn and know, the more we realize we don't know.


The fun part of science is finding things out. Thats just the way i view it.

Yes i am fascinated by physics and math and the stimulating feeling it gives me. Who the heck in this field isn't? What do i care if anyone will understand the whole universe? I get the satisfaction of discovering things i didn't know before and i am fascinated by the fact that humans can communicate even the slightest bit about the universe.

I don't let the fact that no one will understand the universe stop me from acheiving my goal which is just that. It is what drives to learn more. Just because i have a different view does not mean i am not going all in. I have always done everything to be the best and i have always acheived it. But i can accept the fact that there are many thing we will never know. And that is just the way it is

 

[Group_Complex]

if any human thinks they can understand everything that goes on in the universe they're ignorant. we are one of many very insignificant species which our brain cannot process many many things. We know how things work, but that's all. what i mean by understanding the universe is that you have a clear understanding of say, gravity or magnetism. What is that force? not describing what it does, but what IS it? and WHY does these things do what they do. and it is extremely difficult to answer those questions because these things are so bizarre.

don't get me wrong, i WANT to know everything. but here is something to think about. Once someone has discovered EVERYTHING and understand the universe and everything within. That my friend, will be the end of physics. There will be no more reason to go on searching for answers if everything is known.

the fun and beauty of physics IS because it is mysterious. i get shivers down my spine learning about even atoms which is taught to every high school student. My goal is to learn and understand as much as i can.

since you ended your post with a quote, as will i :P lol

"but I don't have to know an answer. I don't feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose which is the way it really is as far as I can tell."
-- Richard P. Feynman

The universe is not MADE for humans to understand.

The point of science is not to understand deep philosophical questions about the universe such as "why does the universe exist", these questions are the realm of philosophy. Science deals with Modelling the natural world. That is the goal, not to become philosophical masters of the cosmos. If the next einstein comes along with a grand unified theory, physics will probably not end (even assuming we can model the fundamental physical interractions) as higher order chaotic systems will still be studied via physics.
This is the main problem with most physics students, they think the point of physics is to discover some deep philosophical meaning to reality, the "reality of time" or some such thing. Even the ones that think the aim of physics is to come up with a simple equation for fundamental physical processes is the purpose of physics are mistaken, physics is a vast subject, with most physicists working in fields that have nothing to do with string theories and multiverses.
The true fundamental science is mathematics, as physics is the attempt to model reality on mathematical objects. The work by Murray Gell-mannin particle physics for instance was an observation that certain symmetries related to mathematical objects called groups, not some deep philosophical musing about the nature of reality. The only way to ever have a true grand unified theory of everything, would be to validate the mathematical universe hypothesis (The universe is just a mathematical structure see Tegmark) as then and only then will any philosophical baggage be removed. This task however seems to me impossible, how to we verify the universe is a mathematical object in an empirical science? We can't, without resorting to philosophy, the very thing we were trying to avoid.

So it seems to me, quite possible that we will model the fundamental physical constituents of reality (Even then we could ask "Is there something more to the foundation of reality that we humans cannot observe?" I mean a Dog will never understand Newtons laws, no matter how many times you throw a ball in the air) but it will not be a philosophical understanding, rather a mathematical one.

 

[jimmyly]

The point of science is not to understand deep philosophical questions about the universe such as "why" questions, these questions are the realm of philosophy. Science deals with Modelling the natural world. That is the goal, not to become philosophical masters of the cosmos. If the next einstein comes along with a grand unified theory, physics will probably not end (even assuming we can model the fundamental physical interractions) as higher order chaotic systems will still be studied via physics.
This is the main problem with most physics students, they think the point of physics is to discover some deep philosophical meaning to reality, the "reality of time" or some such thing. Even the ones that think the aim of physics is to come up with a simple equation for fundamental physical processes is the purpose of physics are mistaken, physics is a vast subject, with most physicists working in fields that have nothing to do with string theories and multiverses.
The true fundamental science is mathematics, as physics is the attempt to model reality on mathematical objects. The work by Murray Gell-mannin particle physics for instance was an observation that certain symmetries related to mathematical objects called groups, not some deep philosophical musing about the nature of reality. The only way to ever have a true grand unified theory of everything, would be to validate the mathematical universe hypothesis (The universe is just a mathematical structure see Tegmark) as then and only then will any philosophical baggage be removed. This task however seems to me impossible, how to we verify the universe is a mathematical object in an empirical science? We can't, without resorting to philosophy, the very thing we were trying to avoid.

So it seems to me, quite possible that we will model the fundamental physical constituents of reality (Even then we could ask "Is there something more to the foundation of reality that we humans cannot observe?" I mean a Dog will never understand Newtons laws, no matter how many times you throw a ball in the air) but it will not be a philosophical understanding, rather a mathematical one.

This is the point i was trying to get across. We can only learn the rules of which govern the universe. String theory and all that does not even cross my mind. That is why i was saying no one will ever understand the universe and everyhing within.

Well said group complex :)

 

[Group_Complex]

This is the point i was trying to get across. We can only learn the rules of which govern the universe. String theory and all that does not even cross my mind. That is why i was saying no one will ever understand the universe and everyhing within.

Well said group complex :)

For all we know our primate cousins understand "reality" or the "true nature of the universe". The nature of the universe and our understanding of it, will always be a philosophical question, open to debate, till there are no humans left to debate it. We learn rules which we observe to govern the universe, but "god" knows what really governs the universe.
That is why I like mathematics, the way I see it, logic governs reality, even if that is my own philosophical outlook. At the end of the day, our search for understanding will be based on logical and mathematical truths.

 

[azizlwl]

For all we know our primate cousins understand "reality" or the "true nature of the universe". The nature of the universe and our understanding of it, will always be a philosophical question, open to debate, till there are no humans left to debate it. We learn rules which we observe to govern the universe, but "god" knows what really governs the universe.
That is why I like mathematics, the way I see it, logic governs reality, even if that is my own philosophical outlook. At the end of the day, our search for understanding will be based on logical and mathematical truths.

If the heavens were indeed controlled by the whims of gods, then these gods themselves seemed under the spell of exact mathematical laws.- Sir Roger Penrose(The Road To Reality)

 

[sandplasma]

You most definitely can. Like everyone says, you need practice and more importantly (in my opinion) a good foundation. All math builds on previous math. You can't expect to understand Calculus without being very good at algebra. etc

 

[SophusLies]

This is the main problem with most physics students, they think the point of physics is to discover some deep philosophical meaning to reality, the "reality of time" or some such thing.

I really don't understand where you got this idea. I can't speak for the majority of physics people I know but I'm positive they aren't in physics to find "deep philosophical meaning to reality." The interests range from cutting technology to applying math in theoretical models.

I don't disagree with the other things you said but I get annoyed when people that speak for physics when they aren't even in it.

 

[wlcgeek]

Yes, no one starts out being "good" at math. The only people who are good got good. Can ANYONE get good? I don't know. To an extent, yes.

http://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

In sports, it's more clear how genes are involved simply because things like muscles, height, and so on are much simpler than the brain. In intellectual pursuits, it's not very clear how genes are involved because we don't understand the brain very well. Who says music is an innate talent? It's in how you practice. Practice the right way and you improve. Practice the wrong way and you won't improve much and you'll pick up bad habits. I play piano and with that, it really takes a lot of willpower to get yourself to practice the right way. If you just let go and play by instinct, it's no good. There have even been times when I have been embarrassed to play the same way in front of my teacher because I knew I was practicing the wrong way out of laziness.

If you have ever worked through Drawing on the Right Side of the Brain, you'll find that many people can actually draw well if they just look at things the right way. Some people might practice drawing their whole life, and someone else with the same ability could outperform them in a matter of days if they just read Drawing on the Right Side of the Brain. I am not sure that is exactly what would happen. Maybe if you just tried to draw by trial and error, eventually, you'd catch on, but it's conceivable that you might never figure it out. One person knows the trick, the other person doesn't and that makes all the difference.

As far as math goes, I have a very mathematical family, so I do think genetics have something to do with my abilities (there might be some environmental influences, but it's not like my parents ever really sat down and taught me math or talked about it with me). However, there are plenty of tricks I picked up, too, just like Drawing on the Right Side of the Brain, where if you know the tricks of the trade, it is a big part of being good at it. Different people may need different tricks. Some people may need to do different things than I did because maybe some things don't come as naturally to them or maybe other things come more naturally than for me.

Interesting.

I appreciate everyone's responses.

It seems like there's at the very least consensus that one can improve their math ability via practice. How specifically "good" you can get seems undecided or unknown. Sounds probably the same with everything else in life, lol.

I only wonder if singing can be improved to a point of sounding acceptable for me?! In that area, I think I'm hopeless in life. :D

...Oh, by the way, you mentioned "tricks."

What sorts of tricks do you use in math...? I'm intrigued by that part of your response, homeomorphic. Can you share more?

 

[Szichedelic]

I will use myself as an example. In high school, I flunked out of trigonometry. I liked the subject but it would take me a week to finish a single-day homework assignment.

However, in CC, I put in a lot more work and I managed to ace the calculus sequence, differential equations, and linear algebra. Since then, I probably put 20+ hours a week on my classes alone. I spend any commuting time, via bus, trying to read my textbooks and do practice problems.

Some have a natural affinity for math, but I have seen a few of my papers falter without a proper work ethic. I do not consider myself naturally able in mathematics. Albeit, through hard work, I believe myself to be on my way to becoming proficient.

 

[chill_factor]

I really don't understand where you got this idea. I can't speak for the majority of physics people I know but I'm positive they aren't in physics to find "deep philosophical meaning to reality." The interests range from cutting technology to applying math in theoretical models.

I don't disagree with the other things you said but I get annoyed when people that speak for physics when they aren't even in it.

my experience as a physics grad student is different. the most esoteric, most "useless" (no industrial application) fields like cosmology and high energy are highly competitive while condensed matter can't get enough people.

 

[H2Bro]

my experience as a physics grad student is different. the most esoteric, most "useless" (no industrial application) fields like cosmology and high energy are highly competitive while condensed matter can't get enough people.

I wonder what it is about fields like cosmology and high energy particle physics that attract so many applicants? It seems like a good chunk of people in physics would, if given the opportunity, have a career studying astrophysics, astronomy, particle physics, cosmology - things of that esoteric nature.

Perhaps people get more meaning out of mathematical understandings than one would guess at given the often pragmatic applications of the field.

 

[homeomorphic]

Maybe you can elaborate on "Drawing on the Right Side of the Brain". Any books you wish to suggest?

You mean drawing? Actually, The Natural Way to Draw is, to my knowledge, the best book on drawing, hands down (but it's for people who are pretty serious about it--I wanted to be an artist, at some point). Drawing on the Right Side of the Brain is sort of the quick and easy route for people who just want to draw competently from life or copy from photographs or other drawings. The "science" in Drawing on the Right Side of the Brain is flawed, BTW, but the drawing part works. Not much to say about them--just check them out from the library.

What sorts of tricks do you use in math...? I'm intrigued by that part of your response, homeomorphic. Can you share more?

There are tricks for learning and a few tricks for problem-solving. One of the big ones for learning is "mental rehearsal". People who perform well in most fields will use some form of mental rehearsal. That allows me to visualize things I wouldn't be able to visualize at first. I just keep trying to visualize it, until I can see it clearly. It's not a matter of pure visualization, but seeing why the pictures make some theorem obvious. When I learn less visual sorts of math, I go through a similar process, just mulling over it until it is obvious to me. You have to go through just the right process to learn how to do this, though. It takes a lot of practice. One trick for problem-solving is to avoid getting stuck. If you find yourself digger the same hole deeper, with no luck, sometimes, it helps to try digging a different hole. This may mean trying a different approach, or it may mean trying to solve a simpler version of the problem or a special case. Another trick is to assume that you have made a mistake somewhere and your job is to find where you made a mistake. It turns out when I am having trouble solving a problem, sometimes, it is because I am making a wrong assumption somewhere, whether it's just misunderstanding the problem or something more substantial. Again, you have to practice doing hard problems before these things really click.