Type of thinking that is conducive to learning mathematics

[Niaboc67]

I have my highs and lows with mathematical understand. From those who are very good at mathematics what trait does you possesses that you think gives you the keys to understanding? I've been trying to become more visual with my learning of mathematics, such as visualizing/conceptualizing the the numbers and abstract concepts. Does this help you? Is it more than just memorizing ideas? what tips and advice would you give someone to become better at learning mathematics?

Thank you

 

[DivergentSpectrum]

Ive found the wonders of computer programming to be very helpful with understanding math concepts. ie: program an algorithm for numerical integration, diff eq solver, vector field visualizer etc. To me it really takes you past the esoteric symbols and let's you get a good idea of the numbers themselves, which has immensely improved my math abilities.

Overall, id have to say half of math is breaking the rules and using intuition, and the other half is being rigorous and following the rules to a t, which means its all about knowing when to do which.

 

[Niaboc67]

Ive found the wonders of computer programming to be very helpful with understanding math concepts. ie: program an algorithm for numerical integration, diff eq solver, vector field visualizer etc. To me it really takes you past the esoteric symbols and let's you get a good idea of the numbers themselves, which has immensely improved my math abilities.

Interesting approach! I just so happen to be studying computer programming. I have yet to fully immerse myself into it yet. I need to get my feet wet. But with what you say makes it all the more encouraging.

 

[Stephen Tashi]

what tips and advice would you give someone to become better at learning mathematics?

Begin by studying elementary logic and basic set theory - how to deal with the quantifiers "for each" and "there exists", DeMorgan's laws and things like that. You can't learn abstract topics merely by vsualizing pictures or forming intuitive concepts. Abstract mathematics involves a legalistic approach. It has much to do with using precise language.

I agree with DivergentSpectrum. Experience in programming is very helpful (-unless you get so interested in programming that you forget about studying math !). Not only does programming force you to think in specific and precise ways, the syntax of programming languages makes certain concepts in mathematical logic clear (such as the "scope" of "variables").

At the same time, you have to use some practical psychology. Legalism is repulsive to most people. There are certainly topics in mathematics that make no sense to me and when I began to read about them. I don't understand why the definitions are written as they are and why there is any interest in the theorems. If you are inclined (or compelled) to study such mathematics, you just have to calm yourself down. Even if you are studying as a hobby or recreation then being calm is helpful.

I'm sure you've had the experience of getting fascinated by some practical question like "How does the mechanism on this toaster work?", "Why doesn't this cabinet door close?", "How did he do that card trick?". This happens when some curious problem arises that is not particularly urgent (not as urgent as "Where did I put my car keys !?"). It's productive to think about mathematical problems with this sort of curiosity and concentration.

Very often when people are trying to study mathematics, they aren't letting questions come into their mnd, they are focused on "the problem in the book". Instead of thinking about the problem they are thinking "I wonder what the answer given in the book is" - or they may be thinking about how long it will take to finish the chapter - or what is the next chapter. Letting a new question come into their mind would just be adding to their burden.

When I see posts on the forum like "Tell me the 50 books I should read in order to become a mathematician", it's clear that somebody is thinking about "being a mathematician" instead of contemplating questions in mathematics. Adults have plans,fears, dreams and daydreams. There is some version of the kid's "I want to be a fireman" or "I might fail the test" going on when we think. To think effectively about math, you need to put such things in the background and actually feel curiosity about a mathematical question.

You have to balance your own questions-and-answers with standard mathematical knowledge. For example, it's common to see posts on the forum by people have formed their own private interpetations and theories of mathematical symbols such as 0.9999... or "dy". You won't learn mathematics just by building private mental sand castles. When you think about a question, you should have some curiosity about how "they" did it. Wondering about how a book does things because you are interested in a question is a very different psychological state that focusing on the goal of "mastering what is in the book".

 

[homeomorphic]

Begin by studying elementary logic and basic set theory - how to deal with the quantifiers "for each" and "there exists", DeMorgan's laws and things like that. You can't learn abstract topics merely by vsualizing pictures or forming intuitive concepts. Abstract mathematics involves a legalistic approach. It has much to do with using precise language.

Completely disagree. Logic and set theory are fine, but depreciating visualization and intuition is not fine. It's partly a matter of style. Not everyone has to do math the same way. There are also different subjects that require different approaches and varying degrees of visualization. Visualization is key for me because visualization enhances memory. One of the reasons why I'm relatively good at math is my interest in the psychology of learning. A massive amount of our brain is involved with processing visual stimulus, so when you visualize math, you can tap into that. Logic and set theory in math are sort of analogous to spelling in writing. Yes, you should learn to spell, but you shouldn't get the impression that's all there is.

Visualization was key to my success at math, and although I am sort of a failure at doing research, I did manage to get a PhD, so I must have been doing something right. To the extent that I was successful at research, proving some new theorems in topological quantum field theory, visualization was essential--so, it wasn't that trying to visualize too much stuff was holding me back, except possibly in terms of spending too much time on the side, trying to develop more intuitive pictures of math (and physics) completely unrelated to my research (that's really more a question of not being willing to specialize). I'm just not good at managing huge tasks like writing dissertations and so on, and I just wasn't that interested in the stuff I was working on, which makes it very hard.

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

By the way, Tao has a lot of advice on how to become a mathematician elsewhere on his website.

Math develops in a lot of different ways. Sometimes visual, sometimes not. A lot of the stuff that may have been discovered by brute force calculations and clever ideas of how to carry them out can later be interpreted more geometrically, which makes it more interesting and easier to remember.

At the same time, you have to use some practical psychology. Legalism is repulsive to most people. There are certainly topics in mathematics that make no sense to me and when I began to read about them. I don't understand why the definitions are written as they are and why there is any interest in the theorems. If you are inclined (or compelled) to study such mathematics, you just have to calm yourself down. Even if you are studying as a hobby or recreation then being calm is helpful.

Being calm is fine, but you might achieve more if you, instead, searched for a different book that presents the material less dogmatically. John Stillwell, Vladimir Arnold, David Bressoud, Tristan Needham, and Cornelius Lanczos are examples of authors who actually explain math, instead of merely presenting it.

Niaboc67, you should read Visual Complex Analysis. If it's too expensive for you, see if you can get it from a library and there are samples of it available online.

http://usf.usfca.edu/vca//

 

[Stephen Tashi]

Completely disagree. Logic and set theory are fine,

I'd say they are more than fine. They are esssential.

but depreciating visualization and intuition is not fine. It's partly a matter of style. Not everyone has to do math the same way. There are also different subjects that require different approaches and varying degrees of visualization. Visualization is key for me because visualization enhances memory.

I agree visualization and intuition are important. But judging from Niaboc67's other posts on the forum, we aren't giving advice to a graduate student. Most students beginning mathematical studies naturally use visualization and intuition. They use that approach so much that they neglect to deal with the formalism. The advice such students need is to pay attention to using precise language and terminology - the legalistic aspects.

By the way, Tao has a lot of advice on how to become a mathematician elsewhere on his website.

I agree with that. And he's a very talented expositor of advanced mathematical topics for to those who can read mathematics at the graduate level.

Being calm is fine, but you might achieve more if you, instead, searched for a different book that presents the material less dogmatically. John Stillwell, Vladimir Arnold, David Bressoud, Tristan Needham, and Cornelius Lanczos are examples of authors who actually explain math, instead of merely presenting it.

You might achieve more and you might achieve less. I agree it's nice to find a good book. However, people (including myself) get into the rather comical procedure of "I can't understand this. Maybe I should try this book. Nope, that didn't work - I'll try another book. Nope, that did work either. Surely at the end of the rainbow there is a book that will explain it to me.!" It's easy to spend hours looking for book reviews and recommendations instead of contemplating a mathematical question.

 

[homeomorphic]

I agree visualization and intuition are important. But judging from Niaboc67's other posts on the forum, we aren't giving advice to a graduate student. Most students beginning mathematical studies naturally use visualization and intuition. They use that approach so much that they neglect to deal with the formalism. The advice such students need is to pay attention to using precise language and terminology - the legalistic aspects.

Maybe that's true for some people, but I don't think it applies to everyone at that level. When I taught low-level classes, like trig/precalc, it seemed that the students had a very limited capacity for visual reasoning. Niaboc67 indicated that he is trying to become more visual, which indicates that his problem IS with becoming better at that and not necessarily with logic, although it could be that he has problems with that, too.

You might achieve more and you might achieve less. I agree it's nice to find a good book. However, people (including myself) get into the rather comical procedure of "I can't understand this. Maybe I should try this book. Nope, that didn't work - I'll try another book. Nope, that did work either. Surely at the end of the rainbow there is a book that will explain it to me.!" It's easy to spend hours looking for book reviews and recommendations instead of contemplating a mathematical question.

For undergraduate level math, I think there are pretty good books that cover most general topics, but maybe you'd be out of luck for certain particular details of some subjects. You can also try asking a professor or figure it out for yourself.

 

[symbolipoint]

Maybe that's true for some people, but I don't think it applies to everyone at that level. When I taught low-level classes, like trig/precalc, it seemed that the students had a very limited capacity for visual reasoning. Niaboc67 indicated that he is trying to become more visual, which indicates that his problem IS with becoming better at that and not necessarily with logic, although it could be that he has problems with that, too.

Do you really mean "very limited capacity for visual reasoning", or actually mean "unwilling to draw pictures"?

 

[homeomorphic]

Do you really mean "very limited capacity for visual reasoning", or actually mean "unwilling to draw pictures"?

Yes, very limited capacity for visual reasoning. That's to be expected, since they haven't had much practice with it. As I've said, I think people who claim not to be visual people are lying because they have similar brains to the rest of us, a large portion of it being devoted to visual processing, and, for example, if you get a room full of random people, you can get them to remember names using visual memory tricks, and so on, which is proof that at least some of the principles behind it apply to most people. It's just that they may not know to use what they have. Applying it to math takes a little bit of skill, and I'm not sure it has that much to do with logic in the formal sense. Students at that stage are most likely just used to memorizing procedures and that sort of thing and not trying to understand why it works. Understanding has more to do with intuition than with logic.

 

[symbolipoint]

I have my highs and lows with mathematical understand. From those who are very good at mathematics what trait does you possesses that you think gives you the keys to understanding? I've been trying to become more visual with my learning of mathematics, such as visualizing/conceptualizing the the numbers and abstract concepts. Does this help you? Is it more than just memorizing ideas? what tips and advice would you give someone to become better at learning mathematics?

Thank you

Draw pictures. Examine what you read, think, and try to form the picture in your mind; and also try to draw it on paper or display-board.
When struggling to understand something you are reading, try to draw a figure to represent it; or try to form a table of data or a chart to represent it.