Higher mathematics learning techniques

[v0id]

What are some methods of training one's mind to absorb and understand rigorous mathematical texts? I have been facing great difficulty as of late in studying fields like abstract algebra, complex analysis and calculus of variations. These are all fields where I am unable to formulate graphical representations of concepts to better understand and retain the information in textbooks. I have been tackling exercises with success, but the problem seems to be more fundamental - it's almost as if there is a "trick" to gleaning connections, that I am unaware of.

My difficulty has arisen quite recently. I have had numerous moments of clarity in the past when my mind has been in a Zen-like state, able to understand the most convoluted collections of glyphs. Am I ignoring the subconscious aspect of reading and focusing too much on the symbolic processing? What are some techniques you use to "get into the zone" where ideas flow into your brain with ease (and stay there for more than a few hours!)?

 

[mathwonk]

there are no tricks, math is just hard.

 

[symbolipoint]

there are no tricks, math is just hard.

Precisely what kind of study schedule did you use while earning your undergraduate and your graduate degrees?

 

[ice109]

there are no tricks, math is just hard.

that's silly. whether you know it or not you employ heuristics to assimilate new knowledge. whether it be diagramming a proof, comparing it something you already know, or getting lots of experts perspectives on the thing.

 

[Asphodel]

there are no tricks, math is just hard.

"There is no royal road to geometry."Really, that applies to most things. If there was an obvious way to do it that was much easier, we would already be using it (although there are plenty of careers to be made in education by finding ways to improve!). Find the study methods that suit you best, develop good habits, and practice, practice, practice. This is the only real thing to do. Don't forget to get help if you're stuck instead of just staring at it getting frustrated - between the professor, any TAs, study groups, random classmates, campus study centers...if you run into a wall, don't stop until you tear it down, but don't sit there doing nothing but telling yourself what a huge wall it is either.

 

[quasar987]

For an abstract theorem, see what the theorem tells you in the simplest cases. Usually, the result it expresses will be intuitively (trivially) true. This helps gaining an intuitive idea of the theorem and then the mind can set that theorem aside and move on.

For instance the implicit fct theorem is this weird thing but see what it says about fonctions in R^2.

 

[will.c]

Precisely. I don't think mathwonk was saying that there's no such thing as study habits, just that there's also not some mathemagical transcendental meditation routine that makes math easy. The whole "zone" thing is a load of crap. The mind has good days, sometimes it has bad days, but you should be working your *** off everyday.

 

[maze]

1) Try to think up concrete examples and special cases.
2) Try to find counterexamples. Even though you will fail, this will help understanding.
3) Relax the conditions a little and try to find counterexamples.
4) Attempt to relate it to other things by using analogies.
5) Work problems, and when you are done stop for a minute and reflect on what you've done. Could you use the same strategy on other problems? What properties of the mathematical objects were key, and which were unneeded? Look back and see if you could have seen the answer at a glance.
6) Read from multiple sources on the same topic.
7) If you've done all this and the intuition is still lacking, put a little question mark (?) in your mind next to the topic, and move on. Sometime later when you have learned more, it may make sense.
8) When you learn something new, always be looking to use what you learned to fill in those old question marks (?) in other topics.

 

[v0id]

Math is indeed hard, mathwonk. I suppose one cannot substitute for intrinsic ability.

Thank you for the excellent advice, maze & quasar987. Considering special cases is invaluable for getting the gist of things, but I find it difficult to reconstruct (or even remember) the more general version in times of need.

 

[Asphodel]

Math is indeed hard, mathwonk. I suppose one cannot substitute for intrinsic ability.

Talent will only get you so far. Eventually, you just have to work really hard.

 

[maze]

Math is indeed hard, mathwonk. I suppose one cannot substitute for intrinsic ability.

Actually you can substitute for intrinsic ability. All you need to do is work hard and work smart with the intent of improving your skills. Soon you will be running circles around people who have "intrinsic ability".

It's not even clear if there is such a thing as intrinsic ability, or if its just that those who seem to display it just worked harder when they were young kids, and they have a headstart on you. There's been research done on this subject, tracking top scientists and so forth, and the evidence seems to suggest the latter.

In any case, taking such an attitude ("I don't understand because I lack natural ability") is an extreme cop out. Maybe its true, maybe it isn't (it probably isnt), but its no excuse for giving up. What you should say is "I don't understand at first, therefore I should keep trying".

 

[Asphodel]

It's not even clear if there is such a thing as intrinsic ability

Which has more aptitude for mathematics, you or a rock? Okay, how about you or a brain damaged mental patient that spends the day drooling on their shirt? We can work our way up, but the point should be obvious - of course there's such a thing as aptitude. The idea that everyone is the same is just plain ridiculous. But no one is so talented that they can get by with never doing any work if they want to learn something difficult like advanced mathematics.

 

[maze]

We can work our way up, but the point should be obvious - of course there's such a thing as aptitude. The idea that everyone is the same is just plain ridiculous.

There is a certain small segment of the population that is incapable, but current research suggests that becoming an expert mathematician, scientist, chess grandmaster, athlete, programmer, and so forth, requires 2 things: 1) approximately 10,000 hours of hard work with the intent of improving (note: playing 10,000 hours of casual games of chess over a lifetime is not the same as intensely studying the game and trying to improve for 10,000 hours over 10 years), and 2) good mentors/coaches/teachers. Thats it. Innate ability doesn't enter into the equation.

 

[mathwonk]

"Precisely what kind of study schedule did you use while earning your undergraduate and your graduate degrees?"

while an undergrad i played a lot of cards, and then when i got more serious, drank myself into a stupor at a local bar while copying out lecture notes.

in grad school i roared around on a motor cycle and partied a lot.

these methods were somewhat flawed as i failed out of school twice.

the second time i was in grad school, bent on succeeding for a change, i got a rug, placed it on the floor, surrounded myself with books, and did not get up for about 10 years, except to bathe, go to school, run for exercise, and shop.

this method, i.e. constant hard work, no tricks, no excuses, seemed better.

 

[ice109]

Which has more aptitude for mathematics, you or a rock? Okay, how about you or a brain damaged mental patient that spends the day drooling on their shirt? We can work our way up, but the point should be obvious - of course there's such a thing as aptitude. The idea that everyone is the same is just plain ridiculous. But no one is so talented that they can get by with never doing any work if they want to learn something difficult like advanced mathematics.

kim peek is practically a drooling idiot but he can play piano concertos perfectly that he's only heard once in his life. so your statements aren't so convincing.

 

[Asphodel]

kim peek is practically a drooling idiot but he can play piano concertos perfectly that he's only heard once in his life. so your statements aren't so convincing.

Can you play piano concertos perfectly that you've only heard once in your life? No?

Anyone claiming there's no such thing as aptitude has been brainwashed by the political correctness propaganda train, and is making claims that have nothing to do with observables. I didn't and still don't think I need to give more elaborate examples to that effect.

 

[maze]

Can you play piano concertos perfectly that you've only heard once in your life? No?

Anyone claiming there's no such thing as aptitude has been brainwashed by the political correctness propaganda train, and is making claims that have nothing to do with observables. I didn't and still don't think I need to give more elaborate examples to that effect.

I used to think that aptitude was a serious factor in becoming great at something, until I saw overwhelming evidence to the contrary. There is no political correctness about it. Read the peer reviewed studies.

 

[The|M|onster]

I believe it was Cicero who said, "constant practice devoted to one subject often outdoes both intelligence and skill."

 

[durt]

I used to think that aptitude was a serious factor in becoming great at something, until I saw overwhelming evidence to the contrary. There is no political correctness about it. Read the peer reviewed studies.

Right... Link, please...

Anyone claiming there's no such thing as aptitude has been brainwashed by the political correctness propaganda train, and is making claims that have nothing to do with observables. I didn't and still don't think I need to give more elaborate examples to that effect.

I agree.

I believe it was Cicero who said, "constant practice devoted to one subject often outdoes both intelligence and skill."

I also agree in general with this sentiment. However, this doesn't mean aptitude isn't a factor. Really, I don't see any purpose for this debate. Both natural aptitude and training are both factors in being good at something. But the only way to improve is to train. Some people may improve faster or to a greater extent than others. Oh well. All you can do is train as effectively as you can.

 

[Oneiromancy]

Can you play piano concertos perfectly that you've only heard once in your life? No?

Anyone claiming there's no such thing as aptitude has been brainwashed by the political correctness propaganda train, and is making claims that have nothing to do with observables. I didn't and still don't think I need to give more elaborate examples to that effect.

You're referring to an extreme case here...We're talking about normal, healthy, human beings here, not mentally retarded people with cretinism.

 

[maze]

Right... Link, please...

Please read
Ericsson, K. A., Krampe, R. Th., & Tesch-Roemer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363-406.

You may also find a wealth of information on the subject by searching scholarly journal databases for the key terms "expert performance", and "deliberate practice". Google scholar turns up a considerable number of the important articles, and you may use the bibliographies to find more.

Ceci, "The nature-nurture debate: The essential readings" has an entire section dedicated to the acquisition of expert performance ability (section VI), and it is filled with a lot of good citations for further reading.

A laymans introduction to the topic can be found here:
http://www.sciam.com/article.cfm?id=the-expert-mind

 

[Asphodel]

I used to think that aptitude was a serious factor in becoming great at something, until I saw overwhelming evidence to the contrary. There is no political correctness about it. Read the peer reviewed studies.

Which was my original point, until we got derailed by the PC wagon. No matter how much of a knack for the subject you do or don't have, everyone who wants to master it eventually has to just buckle down and work hard. Read the thread.

 

[PFStudent]

Hey,

...bent on succeeding for a change, i got a rug, placed it on the floor, surrounded myself with books, and did not get up for about 10 years, except to bathe, go to school, run for exercise, and shop.

this method, i.e. constant hard work, no tricks, no excuses, seemed better.

Awesome. I agree, this is somewhat like the schedule I keep, except I could stand to add the exercise. Thanks mathwonk.

Thanks,

-PFStudent