How to study math to really understand it and have a healthy lifestyle with free time?

Question

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical exercise, I've gained 30 kg, I've spent countless hours studying at night, constantly had sleep deprivation, I've lost my social life, and I got health problems. My grades are quite good, but I feel as though I'm wasting my life.

I love mathematics when it's done my way, but that's hardly ever. I would very much like my career to be centered around mathematics (topology, algebra or something similar). I want to really understand things and I want the proofs to be done in a (reasonably) rigorous way. I've been accused of being a formalist before, but I don't consider myself one at all. However, I am a perfectionist, I admit. For comparison, the answers of Theo, Arturo, Jim Belk, Mariano, etc. are absolutely rigorous enough for me. From my experience, 80% or more mathematics in our school is done in a sketchy, "hmm, probably true" kind of way (just like reading cooking recipes), which bugs the hell out of me. Most classmates adapt to it. I for some reason can't. I don't understand things until I understand them (almost) completely. They learn "how one should do things", but less often do they ask themselves WHY is this correct. I have two friend physicists, who have the exact same problem. One is at the doctorate level, constantly frustrated, while the other abandoned physics altogether after getting a diploma. Apart from one 8, he had a perfect record, only 10s. He said that he doesn't feel he understands physics well enough. From my experience, ALL his classmates understand less than he does, they just go with the flow and accept certain statements as true.

Also, my problem is having weak memory. I forget a lot. Having to study a different subject each day is killing me. If it were up to me, I'd change the way lectures are done. We'd study only ONE subject for a month or two, then have the exams, and the next month or two the next subject. Mixing it up has a terrible effect: I forget things, because I constantly change the topic. That's why I'm always behind schedual. For example, in the second year, during the school year, I understood almost nothing of multivariable calculus, because I had to simultaneously study abstract algebra, topology, computer programming, etc, and couldn't keep up. But then, I devoted the whole three months of summer and got through all 330 pages of theory, understood it very good, including all the proofs ( which I was forgetting along the way), got a 9/10 grade, and had absolutely no vacations (stayed at home), no free time, no sport, no nothing. It was complete crap.

• should I not read the proofs at all?
• should I try less hard, get worse grades and understanding, and 'have a life'?
• abandon the idea of doing math for a living?
• how can I spot the important/illustrative proofs, without studying them completely (it often happens, that I 'get the whole point' only after I've understood the whole proof)

I'm not gifted/bright at all, I'm completely average with a bad memory, but I do have interest in math, good grades, and a horrible lifestyle for the last few years. This question is directed at people who have a career doing mathematics. How did you manage to study everything on time, AND sufficiently rigorous, that you were able to understand it?

ADDITIONS:

I often tend to be the only one to find serious issues in the proofs, in the formulations of theorems, and also in the worked out exercises at classes. Everyone else either understands everything/most, or doesn't understand and also doesn't care for possible issues. Often do I find holes in the proofs and that hypotheses are missing in the theorem. When I present them to the professor, he says that I'm right, and says that I'm very precise. How is this precise, when the theorem doesn't hold in it's current state. Are we even supposed to understand proofs. Are the proofs actually really just sketches? How on earth is one then supposed to be able to discover mathematical truths? Is the study of mathematics just one big joke and you're not supposed to take it too seriously?

NOTE:

I have a bunch of sports I like and used to do. Also, I had a perfectly good social life before, so you don't need to give advice regarding that. I don't socialize and do sport because digesting proofs and trying to understand the ideas behind it all eats up all my time. If I go hiking, it will take away 2 days, 1 to actually walk + 1 to rest and regenerate. If I go train MMA, I won't be focused for the whole day. I can't just switch from boxing to diagram chasing in a moment. Also, I can't just study for half an hour. The way I study is I open the book, search up what I already know but forgot from the previous day, and then go from theorem to theorem, from proof to proof, correcting mistakes, adding clarifications, etc. etc. Also, I have a bad habit of having difficulty starting things, but when I do start 'my engine', I have difficulty stopping, especially if it's going good. That's why I unintentionally spend an hour or two before studying just doing the most irrelevant stuff, just to avoid study. This happens especially when I've had more math than I can shove down my throat, I have mental preparations to begin studying. But when my engine does start and studying goes well (proven a lot, understood a lot), it's hard for me to stop, so I often stay late at night, up to 4 a.m., 5 a.m., 6 a.m. When the day of the exam arrives, I don't go to sleep at all, and the night and day are reversed. I go to sleep at 13h, and wake at 21h... I know it's not good but I can't seem to break this habit. If I'm useless through the whole day, I feel a need (guilty conscience) to do at least smth. useful before I go to sleep. I know this isn't supposed to happen if one loves mathematics, but when it's 'forced upon you' what and how much and in what amount of time you have to study, you start being put off by math. It stops being enjoyment/fun and becomes hard work that just needs to be done.

Answer 1

In my view the central question that you should ask yourself is what is the end goal of your studies. As an example, American college life as depicted in film is hedonistic and certainly not centered on actual studies. Your example is the complete opposite - you describe yourself as an ascetic devoted to scholarship.

Many people consider it important to lead a balanced life. If such a person were confronted with your situation, they might look for some compromise, for example investing fewer time on studies in return for lower grades. If things don't work out, they might consider opting out of the entire enterprise. Your viewpoint might be different - for you the most important dimension is intellectual growth, and you are ready to sacrifice all for its sake.

It has been mentioned in another answer that leading a healthy lifestyle might contribute to your studies. People tend to "burn out" if they work too hard. I have known such people, and they had to periodically "cool off" in some far-off place. On the contrary, non-curricular activities can be invigorating and refreshing.

Another, similar aspect is that of "being busy". Some people find that by multitasking they become more productive in which of their individual "fronts". But that style of life is not for every one.

Returning to my original point, what do you expect to accomplish by being successful in school? Are you aiming at an academic career? Professional career? In North America higher education has become a rite of passage, which many graduates find very problematic for the cost it incurs. For them the issue is often economical - education is expensive in North America.

You might find out that having completed your studies, you must turn your life to some very different track. You may come to realize that you have wasted some best years of your life by studying hard to the exclusion of everything else, an effort which would eventually lead you nowhere. This is the worst-case scenario.

More concretely, I suggest that you plan ahead and consider whether the cost is worth it. That requires both an earnest assessment of your own worth, and some speculation of the future job market. You should also estimate how important you are going to consider these present studies in your future - both from the economical and the "cultural" perspective.

This all might sound discouraging, but your situation as your describe it is quite miserable. Not only are you not satisfied with it, but it also looks problematic for an outside observer. However, I suspect that you're exaggerating, viewing the situation from a romantic, heroic perspective. It's best therefore to talk to people who know you personally.

Even better, talk to people who're older than you and in the next stage of "life". They have a wider perspective on your situation, which they of their acquaintances have just still vividly recall. However, even their recommendations must be taken with a grain of salt, since their present worries are only part of the larger picture, the all-encompassing "life".

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Finally, a few words more pertinent to the subject at hand.

First, learning strategy. I think the best way to learn is to solve challenging exercises. The advice given here, trying to "reconstruct" the textbook before reading it, seems very time consuming, and in my view, concentrating the effort at the wrong place

The same goes for memorizing theorems - sometimes one can only really "understand" the proof of a theorem by studying a more advanced topic. Even the researcher who originally came out with the proof probably didn't "really" understand it until a larger perspective was developed.

Memorizing theorems is not your choice but rather a necessity. I always disliked regurgitation and it is regrettable that this is forced unto you. I'm glad that my school would instead give us actual problems to solve - that's much closer to research anyway. Since you have to go through this lamentable process, try to come up with a method of memorization which has other benefits as well - perhaps aim at a better understanding of "what is going on" rather than the actual steps themselves. This is an important skill.

Second, one of the answers suggests trying to deduce as many theorems as possible as the "mathematical" thing that ought to be done after seeing a definition. I would suggest rather the opposite - first find out what the definition entails, and then try to understand why the concept was defined in the first place, and why in that particular way.

It is common in mathematics to start studying a subject with a long list of "important definitions", which have no import at all at that stage. You will have understood the subject when you can explain where these definitions are coming from, what objects they describe; and when you can "feel" these objects intuitively. This is a far cry from being able to deduce some facts that follow more-or-less directly from the definitions.

Answer 2

Let me tell you that the only thing that I have been doing for the last four years of my life is mathematics. I have enjoyed the experience thoroughly but I have also had points where I was somewhat unsure as to how to approach my learning. I think that there is no one rule that works for everyone; however, let me answer some of your questions. I hope that I can help:

Question: How to study mathematics the right way?

Answer: I think that the best way to study mathematics is as follows. Let us assume that you have already chosen a mathematics book on a subject that you are really interested to learn. When you read the book, aim to actively think about the subject matter in different ways. For example, if a definition is presented, spend at least 30 minutes to think about the definition. If you are studying a book on linear algebra and the definition of a "nilpotent operator" is presented, you should try to discover some basic properties about nilpotent operators on your own without reading further. This can be difficult at first but ultimately an ability to do this effectively with as many definitions as possible is important in research mathematics.

Let us take the following example in elementary group theory. The author presents the definition of a maximal subgroup of a finite group $G$: a subgroup $M$ of $G$ is said to be a maximal subgroup if $M$ is a proper subgroup of $G$ and if there are no proper subgroups of $G$ strictly containing $M$. You should try to take the following steps:

(1) Find examples of maximal subgroups in finite groups and begin with the most trivial examples! For example, the trivial group can have no maximal subgroup. If you understand this, you have grasped one point of the definition. The next step is to consider the simplest cyclic groups. What are the maximal subgroup(s) of the cyclic group of order 2? What are the maximal subgroup(s) of the cyclic group of order 4? Think about basic examples such as this one. When you are ready, try to formulate a general theorem on your own which concerns maximal subgroups of a cyclic group of order $n$. You should arrive at the theorem that a subgroup $H$ of a cyclic group $G$ is maximal if and only if the number $\frac{\left|G\right|}{\left|H\right|}$ is prime.

Continue to find other examples of maximal subgroups in a finite group. The next step is to consider the Klein 4-group and the permutation groups of low orders. I hope at this point you are really fascinated by the concept of a maximal subgroup. At first, the definition might seem like something arbitrary; however, now that you have thought about it, you have started to gain a sense of "ownership" over the definition.

(2) It is now time to formulate and prove some theorems about maximal subgroups. Again, think of the easiest examples. One thing that can be discouraging for a beginner is to not be able to answer a question that looks easy over a long period of time. What is a good example of an easy theorem? You can study those finite groups which have exactly one maximal subgroup. What can you deduce about such a group? If you find that you are stuck, try to work back to the examples of maximal subgroups that you devised earlier. In fact, this question can be answered quite satisfactorily; a finite group with a unique maximal subgroup is cyclic of prime power order.

(3) The next step is to conjecture some more properties about maximal subgroups based on the examples you devised in (1). For example, you worked out that if $H$ is a maximal subgroup of a finite cyclic group $G$, then $\frac{\left|G\right|}{\left|H\right|}$ is a prime number. Is this true for all groups $G$? Can you think of groups $G$ for which this is true?

Notice how one can deconstruct a simple definition to arrive at a host of interesting questions? This is what a mathematician does all the time and is a very important skill. It might seem difficult at first but doing this will make mathematics all the more exciting and will give you a sense of "ownership" over the content. You worked out this piece of mathematics. This is the way I learn mathematics and I can tell you with confidence that if you practice this, it will soon become the norm.

What do you do after you look at the definition and have thought about it extensively? You continue reading the text. There is a good chance that you will notice the author stating some of the results that you discovered on your own. With luck, there will be results that the author has not stated. If this is the case, it could be a good idea to ask (on this website, for example) about the originality of the result.

However, you will encounter theorems concerning the definitions that you simply did not think about. You should resist the temptation to see the proofs of these theorems and rather you should try to prove these theorems on your own. Think about the theorem for at least a few hours before giving up. Note that theorems with quite short proofs can require highly original ideas and therefore you should not pressure yourself to prove the theorem in a small amount of time.

At first, you will take a long time to prove some theorems. There will be routine theorems and these should be proven fairly quickly. But there will also be difficult theorems. As you become experienced, your thinking will be faster and these theorems will come more easily to you. However, you should not expect this to be the case initially.

For example, you might encounter the following theorem in linear algebra: if $N$ is a nilpotent linear transformation from a vector space $V$ to itself and if the dimension of $V$ is $n$, then $N^n=0$. Working out how to prove this theorem on your own is a very valuable and rewarding experience. If you have not seen it already, I suggest that you try to prove it. It is not too difficult, however.

Question: How to avoid forgetting mathematics?

Answer: I used to forget mathematics too when I learnt it. I have talked to various mathematicians about this and they have said exactly the same thing. The point is that you just have to accept from the start that you will forget what you learn. However, there are ways to ensure that you keep this to a minimum.

For example, the best way to not worry too much about forgetting mathematics is to work out the mathematics on your own. For example, consider the steps that I suggested in the previous question. Even if you do this, you can still forget the mathematics, especially if the result in question was fairly easy to prove. (Note, however, that if the result is hard to prove, and you spend, let us assume, 10 hours to prove it, then you will probably never forget it for the rest of your life.)

The best method to take is to write down all the mathematics that you learn. Take copious notes. For example, when I read Walter Rudin's "Real and Complex Analysis" last year, I took down 3 entire books of notes. In fact, I wrote down 600 pages of mathematics when I only read 315 pages!

Write down every definition, every theorem, and every proof. The definitions and theorems should be produced verbatim from the book since it is important to ensure that your understanding of the rigor is correct. However, the proofs should be written in your own words.

Question: How to have a healthy lifestyle?

Answer: I am afraid I really do not have a good answer for this. In the four years that I have been studying mathematics, I have certainly not done anything else. Therefore, I cannot really give advice on how to manage one's time. If you are a serious student in mathematics, you will find yourself spending virtually your entire day doing the subject. This is inevitable. For example, I set myself goals every day of how much mathematics I wish to do and usually I end up doing mathematics non-stop. Nonetheless, I really enjoy this and I would not wish to have it any other way.

But I can offer one small piece of advice: try to wake up early, let us assume, at 6:00 AM. However, do ensure that you sleep for at least 8 hours; therefore, go to bed at 9:00 PM. Sleep is one of the most important points when it comes to studying. Over many years of doing mathematics, I have found that I am most productive and energetic before 12:00. If you can finish off most of your work before 12:00, then you will be in a really good position to do well each day. Also, try to avoid eating big meals. Big meals often cause you to lose your concentration and this can, in turn, lead to several wasted hours.

I think the most important point when you set out to achieve any goal in your life is to take it day by day, hour by hour, even minute by minute. Often you can complicate goals too much by thinking of what you would like to do over the next 1 year or even one month. If you work hard each and every day and set realistic goals, then anything should be possible.

I hope that I have helped! (I hope that my usage of bold text is not considered offensive; I simply used it to highlight some of the key points in my answer.)

Answer 3

Regarding the "and have a healthy lifestyle" thing. Well, you also have to learn when to stop.

Sometimes you figure out you don't understand something and you don't have time right then to understand it. Try to "box" that as much as possible. Figure out the general form of the kind of thing you don't understand. How does it function? What context is this idea used in? What kinds of "inputs" does it have? Does something appear as if by magic? What? Have you ever seen anything like that before? If you keep these vague ideas in mind, you may very well figure it out in your sleep, in a conversation with someone, maybe weeks later, maybe years. It depends on the particular thing.

Sometimes you don't understand something because that thing is complete nonsense. Profs sometimes say nonsense -- they're human beings and make mistakes. Books make mistakes. I remember spending a lot of time trying to complete a proof for a homework problem and everything I tried failed. An hour before the homework was due I was talking with someone else in class. He showed me an example he cooked up to demonstrate the theorem and I reinterpreted it as an example that disproved the theorem. These things happen. Similarly, plenty of textbooks have subtle "lazy" errors in them. If you're not particularly confident in yourself you may spend hours being frustrated on such a problem. Talk with people.

Answer 4

Some very basic non mathematical advice and I'm sorry if I sound like your mother. If you feel like your memory is bad and you're not finding enough time to socialise, perhaps you're not finding enough time to eat well. Eating plenty of fresh fruit, fresh vegetables, fresh fish, olive oil and cereals will give your body the building blocks to do its best . Oily fish in particular are known to be good for the brain. http://www.newscientist.com/article/mg20827801.300-mental-muscle-six-ways-to-boost-your-brain.html

I find that the time I spend cooking / washing up is pleasurable mentally relaxing down time during which some of my best ideas come. Perhaps you could combine this with a social aspect and invite people round to tea if you're cooking something nice. Avoid the alcohol that usually goes with these situations if you're intending to get back to work after.

Answer 5

Most people probably won't like this answer, but mathematics is a field where there's an unstable separation between those of genius caliber understanding and those who are just able to get by through dogged hard work. Way too many people want to do proofs and aesthetically pleasing artful mathematics for a career who are in the dogged-hard-work category. I am speaking as someone who has worked myself to death over the past 3 years to hack it in an Ivy League Ph.D. program in applied mathematics. Next to my peers, the only advantage I have is that I am able to work much harder, and to some extent I am much better at writing software. In terms of mathematical prowess, they all dominate me.

If, as you admit, you are average with a bad memory (aside from your obviously above average tolerance for difficult technical work), then you need to consider that an actual career in pure mathematics is not right for you. I want to be careful to avoid other-optimizing so please take my advice with a grain of salt. It may not be right for you, and surely all of the other commenters have insightful advice as well. But one thing that I think will never work for you is to just "try to exercise, eat right, and have a balanced life." Whatever others say, this will not happen for a pure mathematician who has really good taste in the aesthetic beauty of results, unless that mathematician really is at the genius level.

You have a limited talent supply and a limited time budget. Your personal forecast that you'll enjoy a career in abstract mathematics is almost surely incorrect; you seem to undervalue important things like salary, competitiveness for tenure, geographic preferences, etc.

For example, I have a close friend who studied very pure aspects of cryptographic number theory. He did two post-docs and earned practically no money at all, sacrificed personal relationships to try to get tenure track faculty positions, and ultimately found no jobs doing pure math. He took a job as a programmer for a company that makes cryptographic software. He thought that at least some of his time would go to researching new asbtract ideas in cryptography, but it turned out not to be true. Instead, he writes Java programs most of the time, learns about new applied cryptographic research, and writes very little (though he still dabbles in research in personal time and is, in my view, far more educated about cryptographic research than most people who currently publish in that field).

Is he unhappy in this situation? No! Actually, he discovered that to do software design properly, virtually everything is all about understanding the right abstraction, the right data encapsulation, the right design pattern, and this not only has great mathematical aesthetic value, but also delivers a better product to a client. After acclimating to professional software development, he now sees all sorts of parallels between his former work coming up with abstract math ideas and his current job coming up with abstract software solutions. His skills set now has a far higher economic demand, he isn't pressured to compete for tenured positions, and he's able to keep a very healthy work/life balance because of his company's regular work schedule.

I would say that, just as so many small businesses fail, far too many bright-eyed grad students see themselves as the next Godel, gung-ho for tenured positions and "living the life of the mind." They are especially prone to do what you are doing and to let the rest of their lives deteriorate in the hopes of being able to pursue what they currently (probably mistakenly) think is their own preference for abstract beauty that can only be satiated (also a mistake) by generalized math. Many more of these people should own up to the fact that they are not talented enough, and that universities that have to dedicate an ever slimming number of resources to hiring the best tenured faculty shouldn't really hire them.

Here are a few links to consider:

• The Disposable Academic
• Why post-docs are replacing principle investigators

The economics also matter. Tenure is greatly diminishing in many parts of academia, and math faculty are especially notorious because pure math doesn't bring in grant money the way applied projects do. With the advent of online courses and open courseware, and sites like Stack Exchange, the need for highly specialized math teachers is diminishing at the university level. You should expect competition for tenured jobs to tighten, and that if you want a tenured job you'll have to go anywhere that offers them, even if this is a small regional university that is nowhere near any major city, has no real cultural atmosphere, and doesn't attract gifted students. It would be a big mistake to fail to take this into account.

My advice to you is this. Think hard about what it is specifically that you enjoy about mathematics. If you like the abstractions and geometrical thinking that are often part of advanced analysis and topology, then there are many applied mathematics / applied physics / engineering career routes that will offer you the chance to explore math questions, but will also put that geometrical abstraction ability to work writing software to solve actual problems. Your familiarity with pure mathematics may give you a career edge if you switch to a field like this. You might be situated to compete more effectively for grants and faculty jobs if that is what you want, and to the extent that you master programming skills, you'll have marketable skills to get different jobs if the need arises.

If you prefer the more abstract thinking that often accompanies algebra and number theory (that is, if you are a "problem solver" type of pure mathematician according to Timothy Gowers' definition (see here for my take on that if interested)), then I think you will find a lot to enjoy about software design. You may be better served by focusing on abstract problems in computer science and software engineering.

If you read a good math history book (e.g. Stillwell), you'll notice that (a) most good abstract math begins by being some sort of ad hoc, "it's probably true but I can't see the details" intuition anyway and it only gets refined later; and (b) most awesome stuff invented in mathematics was not invented by people who thought that mathematics was the way they needed to earn a living. People have been driving themselves mad over solving math problems for millennia, staying up late into the night, leading destructive romantic lives, falling into ill health. If you really love math, you'll never be happy doing it in the half-assed way that a healthy work-life balance requires, and very few people are truly capable of sustaining a career like that. Most ultimately stop trying hard on the math part and become dissatisfied with their careers.

Earning a living by being a pure mathematician is a very modern concept that arose largely because of the implications of Lebesgue integration in analysis and computability theory in computer science. And now that we have enough of a handle on those fields and their subsequent children, there just isn't enough stuff to support a lot of career mathematicians. Almost surely, significant mathematical advances in the next 50 years are going to come from highly intelligent, dedicated hobbyists, who solve problems at places like Stack Exchange or polymath.

And there's no reason why you can't find some niche problems that you like to work on, do so in your free time, and meanwhile have a fulfilling and economically sensible career that affords you a more comfortable life. For as obviously smart as you are, it would not be wise to fail to consider this sort of thing in your youth. Many more math students should do so as well.

In fact, the really egregiously unfair underfunding of students and inflation of post-doc positions largely comes about because naive youngsters who think they will automatically get tenure if they just try hard, and who think that nerdy love for aesthetic science is a good thing to base a career choice on, seem to unquestioningly accept underpaid and under-insured academic positions with no question. Trust me, you don't want to just be another one of those folks.

Answer 6

On studying math:

Your time on this rock is finite, the amount of mathematical knowledge is infinite. You must choose wisely about what you want to spend your time learning. Decide if you want to be a "jack of all, master of none" or "a master of one, jack of none" type.

Personally, I'd rather know a lot about everything than everything about one thing, thus I'd say don't waste your time learning every single detail. Appreciate the high level material, move on to the next field until you run out of fields. Then, work your way down to lower level material as your time allows.

Your learning doesn't stop once you complete your degree(s). Learn to pace yourself.

On the healthy lifestyle side of things:

The mind can only be as sharp as the body. Make the time to exercise, eat right and get enough sleep. You will find that you think more clearly, you retain information more effectively and that you are happier.

Balance is critically important. You get but one life to live and there is far more to it than mathematics. Take the time to explore other interests, discover new ones and become more well rounded. The greater your overall knowledge, the better you will be at math.

Live life. Socialize, fall in love, run a marathon, get into a fight, go to the ballet, paint a masterpiece, go fishing, explore the world; do the things that make you more than just a mathematician, do the things that make you a person.

Answer 7

I can see that this question is a couple of month old, but I would like to add some remarks:

1) Most research mathematicians have a better memory and are quicker than what you describe. There are notable exceptions, but you have to understand that it will be hard to compete. You will always have to work harder than most of your peers. If it takes you three times as long to correct exams, then this time will be missing from your research even though you might be just as talented for actual research. On the other hand, don't trust your fellow students when they just say that they understand things and are quick. In many places, it is cool to claim to have aced the exam with little study time. In the long run, you might overtake some of the people who know how to learn just for an exam.

2) Of your peer group, a very small percentage will become researchers. There is no point in comparing yourself to people who efficiently pass the exams if you want to become a researcher. Seek out good, ambitious students and socialise with them. If they are quicker and have a better memory than you, then ask them what is wrong with Lemma 3.4 whose proof seems somehow strange to you. There is no point in finding all stupid errors yourself. Ask your peers, ask your professors, ask here. You are wasting time if it takes you three hours to find out that the professor wrote "c" instead of "e".

3) If you concentrate too much on details, you have to train summaries. Can you explain to a very talented beginner student what they will learn in linear algebra and analysis? In 10 sentences? In a couple of minutes? In a couple of hours? Without paper? When I need a result from a lecture that I heard as a student 15 years ago, I don't need to remember the conditions in the theorem. I need to realize that this theorem is probably applicable to my problem, in which lecture or book I saw it and then I can look it up to check whether there was some technical condition I forgot. To do so, I have to remember the gist of the theorem and the proof, not the details. Also, if you don't understand something during learning, preliminarily accept the result, continue and return to the result later, don't brood on one thing indefinitely.

4) Usually, the gist is something professors like to hear during an oral exam. They will check the details here or there, but they don't need to hear the gritty details all the time. Are you sure that you are acutally speaking at the expected level of detail during your oral exams? Or are you just assuming that the professor wants to hear all details and start right away at the epsilon level? Have you ever tried to sit in on other students exams?

5) Seek out younger students and help them preparing for their exams (or answer questions here). Helping others is the best method to keep your acquired knowledge fresh. This will not be wasted time.

6) You should absolutely not sacrifice your physical and mental health. Sleep, food, exercice, social life and hobbies are important and should not be neglected for an extended time period. Is it not possible to just take fewer lectures per semester? Who will care later if it takes you a year longer to finish? Something has to give, and it seems to me that the easiest thing for you is to just spread the work over more time. (And yes, I do realise that even without tuition there are high opportunity costs, but you seem to need more time now.)

Answer 8

Being in good physical shape, makes your head clearer. Besides, you only need one hour of exercise a day to keep in decent shape. Do some multitasking! Walking and thinking can be done at the same time, you train your memory, AND your stamina.

Solving math problems in your head is a good mental exercise.

Answer 9

I am an undergraduate at an American university going through almost exactly what you're describing... the lack of sleep, lack of social life, weak memory for proofs, and a perfectionist insistence that mathematics be presented "my way." Like I say, I'm currently going through this, so I can't offer any answers. I do, however, have some suggestions from experience.

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Recently I found myself in the unfortunate situation of having to memorize many proofs the day before an exam. Suddenly, it no longer mattered if I knew only a few proofs in great detail. What I needed was to know all of the proofs, but only in enough detail to warrant sufficient partial credit. To do this, I skimmed the proofs in my textbook one by one, writing little summaries of each in my own words.

My point is that this (for me) was a very effective method of grasping the big ideas of the proofs without getting hung up on the details. In writing my own summaries, I was also able to boil down entire proofs to a couple of sentences, which then served as mnemonics for memorization.

But as for the questions you actually asked...

Should you skip reading the proofs? Ideally, you'd read and understand all of them, but if you're crunched for time (as you seem to be), then you have to be efficient. Ryan Budney is right: you have to learn when to stop. Learn what you think is relevant to doing well in the class. Then, when the course is over, you can take the time to understand the details or less-important proofs or whatever you want, should you so desire.

Should you try less hard, get worse grades, but "have a life"? I don't think anyone can answer that but you, I'm afraid.

I will say, though, that efficiency really matters, and that you might be able to find ways to balance academics with a social life if you look for them. You know, somehow we're all pretty efficient when exam time comes around, managing to cram large amounts of information in a very short amount of time. We have no choice but to be efficient. So while I'm not saying that you should treat every day like it's the day before an exam, I do think that you can find ways of increasing efficiency if you look for them.

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I should point out that all of this is meant to be practical advice rather than sage advice. For sage advice, I also recommend Terrence Tao's career advice, as well as talking to your professors and advisers.

Finally, I should mention that it is my understanding that -- although I am by no means a professional mathematician just yet -- that at the end of the day, discipline and hard work matters just as much as natural talent, if not more.

So if you're worried about being able to produce research-level math, then my advice would be to stop worrying about it. If you haven't actually tried your hand at research yet, then there's no reason to worry yourself about it prematurely. At least, this is what my adviser told me when I presented him with these concerns last year, and really, it's been some of the best advice I've ever received.

Answer 10

I think Terence Tao's Career advice can answer your question. I would strongly recommend you to read it.

Edit: And also Kevin Houston's How to Think Like a Mathematician: A Companion to Undergraduate Mathematics.

Answer 11

I suggest transferring to a program where you can make your interests coincide as much as possible with the things you're required to study.

Mid-way through my 1st year as an undergraduate I stumbled upon the honours mathematics program at the University of Alberta and I was pretty much hooked. The honours program emphasized rigor, understanding, technique, visualization, precision, basically just a really solid foundation.

In my 2nd year as an undergraduate I had a (required) rather unfortunate introductory differential equations course where it was all crank-the-formula. Proofs and ideas were nowhere to be seen. That course was quite frustrating for me -- it seemed like such a waste of an opportunity to start connecting the various threads we had been developing in analysis, linear algebra, algebra and so on.

I found as an undergraduate I usually had plenty of time to do everything I was required to do. There were moments when I got in situations that were close to being over my head but it all worked out for the best. If you never push yourself too hard you'll never know what "too hard" is. So it's a good thing to discover. I think it helps when the things you have to do are the things you want to do. If they're not, you can end up wasting time being bored out of your skull. It's a good lesson to learn how to accomplish boring things, but hopefully there's not too many of them in your undergraduate education!

Answer 12

I was (and am still to an extent) going through much a similar phase some time back. I am not particularly good in all the main subject areas of mathematics and am precise about details myself and there occur times when it all sorts to overwhelm me a little. At such times I either indulge in fun-maths, and just try to prove results for fun which attract me, no matter how much time it takes, or how trivial they seem. This keeps me attached to mathematics while also relaxing me. Also I take time out daily for non-mathematical activities because if I dont, too much mental activity invariably gives me a headache.

I think doing maths is much like playing music. It is hard work, but occasionally you can play whatever tunes relax you. All said and done, I think it is important to remember why exactly we do maths: because it is fun!

Added after seeing the comment:

I see. I can sort of relate to my graduate days with that. I got through them somehow with a lot of angst, and what I learnt from that was this: Its important that you study the proper way and that proper way is unique to everyone. In my second semester I remember really struggling through Complex Analysis by Ahlfors (I still am a little apprehensive towards it) and the reason was that that book was not geared towards my way of studying, and there was no time to painstakingly give arguments for everything "assumed to be clear" in the book. Later on, I read another book on Complex Analysis(Brown & Churchill) and what I know of the subject is largely due to that. This is because the second book was more geared to my internal understanding process then Ahlfors. Perhaps you too will go a lot of quicker with the grades if you read from books that appeal to you intuitively and not from prescribed ones.

Answer 13

You might consider distinguishing the understanding of mathematics from the requirements of your classes. If you need to memorize a proof for a test, in order to remain in school, then do it... but don't confuse it with understanding, or with working out proofs for yourself. You might find that breaking a proof into the parts you do understand and the parts you don't understand will simplify the process of coping with the proof.

Who knows? Maybe having memorized a proof you don't understand, you'll find that you can think about it while you're doing something else, and perhaps understand it in a flash of insight.

Answer 14

This answer will attempt to only address the first part of your question. When I was doing undergrad work, I gained a lot of weight since my main way of doing my homework was just sitting down and eating chips or something while putting my nose to the grindstone. This, in combination with needing to constantly study, was really bad for my body and also caused some anxiety problems down the line. Around my 4th year, I started making daily to-do lists which included little bits and pieces of things which were not math-y: I found short (10-15 minute) exercise videos on youtube that I knew I had time to commit to, and I did that "100 Push up Challenge" which you can probably find via google (I didn't quite get there, but I had a lot of fun along the way!).

To this day, I set aside time for at least 20 minutes of exercise each night (you'd be surprised at how focused you are afterwards) and I feel significantly better, physically. Once you've been doing it for a month or so, it just becomes natural.

As far as the social problem, different people do different things. I ride my bike to school, so I joined a cyclist group. I also found a number of other social clubs in the city (Chicago at the time) that did things I liked. I was surprised at the number of people just looking for other people to talk to, and not all of it was science-y!

It may be the case that this does not work for you, but I wanted to share my experience just in case someone found it even a little bit helpful.

Answer 15

If you choose to study mathematics so hard, as you describe it, then it means that you should like it, and I guess you do. Still, saying that it eats up all your time worries me. Math should be a pleasure; at least that's how it is for me. Math shouldn't take all your time, because your brain needs to rest for you to process things more easily. Here are a few tips:

• when studying, choose a degree of detail or depth you wish to go through. Do not work out all the proofs from a book, when studying it. Choose what really interests you and what you need for your course. Usually a course does not cover an entire book, and for getting good grades you don't need to now way more than what has been taught in the course. Some things will get clear only with time and experience; you will learn them for the exam, but you'll understand the whole picture in a larger period of time, maybe years.

• learn to relax everyday. Maybe I'm a lazy type, but I always find time for a walk, a bike ride, for playing the piano, for watching a movie. For example, I relax when solving problems on this site, or on my own, problems which do not have anything to do with what I'm studying at the moment. Take at least 8 hours of sleep per night. When relaxing you leave the brain a chance to put all the things in order. Many mathematicians had revealing ideas while doing ordinary things. A walk in the park can help you understand a key point in a proof, or a solution to a problem might pop up when doing some sport or some chores around the house (it happened to me more than once; the funniest one was that I solved a problem given to a team selection test for the IMO in my head, without any pen and paper while cutting the grass in the field with my father and grandfather).

• usually you can focus better if you have a better goal than 'finishing the book'. For example, take an article in the field you're studying (maybe a teacher can help you with that) and try and understand that article in detail. Study only the theorems and proofs which are related to that. Mathematics has developed enormously in the past years. Trying to keep the pace with everything is impossible. Focusing on a narrower scope is usually easier, and in research this is really needed.

• do not ever worry about memorizing everything. You will forget many things no matter how many times you learn them, but the essential thing to do is to remember where to look for the things you forgot. For example: theorem X with examples and counterexamples is presented in book Y, subject Z can be found in the book T, and so on. Try and split your proofs into steps you can remember. Do not memorize calculations. Remember only key points, and trust yourself that you can fill in the blanks.

• find time to spend with friends or colleagues. Having someone to share an idea with, even a mathematical one, can be of great help.

• find someone you can tutor ( at highschool or university level, in a lower year ). This can be of great help financially and you'll notice how good you understand things when you try and explain them to someone who doesn't know them at all. This has been of great help to me.

Good luck.

Answer 16

For the weight gain part: Read some Arthur De Vany's writings, and check out his diet. People eat constantly and expect not to gain any weight. De Vany suggests, some days, after a healthy, big breakfast it's okay not to eat anything, and going to sleep hungry. "Going to sleep hungry" part gets big objections from people, but our bodies have not evolved from their hunter gatherer structure just yet. When humans hunted, they could go days without eating much. Our metabolism is geared towards feasting (overeating) after a good catch, and going around hungry until that catch. Constant eating is not our way. Try this and you will happier, your body when it is hungry will be relieved even, because there will be one less thing it needs to work on. On days when you are not eating (after breakfast), it is also advised to set that day aside for physical activity, walking, etc. (yes, while hungry), but your mind will rest in the meantime.

Answer 17

Set your priorities. Get a calendar. Put exercise in it for a half hour 3 times a week. This is better than nothing. Don't do anything that's going to absolutely wear you out. Schedule time to hang out with people. Schedule other priorities like sleep. Then, whatever time is left over, which would probably be quite a bit still, do math. If you still feel the same way, schedule more exercise/sleep/hang out time/whatever.

Lots of people forget math. I wanted to study algebra stuff. I had Real Analysis 1 and 2 (graduate level) one year and spent so much time on those that I had no time for algebra. I couldn't remember the algebra stuff that was actually important. The real analysis isn't even important to me. I should have prioritized better and spent less time on real analysis and more time on algebraic stuff.

I spent so much time away from my family that year also, which was terrible. I now basically work 8-5 and go home. And, maybe Saturday I work for a few hours. Sometimes, I try to get in some more work. But, my family is more important than this math so I don't care if it takes longer to graduate or I don't do quite as well.

Answer 18

"Also, my problem is having weak memory. I forget a lot."

Have you tried using a flashcards program? There is a program where you can add questions. If you answered a question correctly, the program will ask you the question again in 2 days. If you answer correctly in 2 days, it will ask you again in 4 days. After that, the interval will grow to 8, 16, and so on. I found it helpful. You can also set a custom interval. The program accepts mathematical symbols.

Also, don't add EVERYTHING you want to remember to the program. You'll remember a lot of things anyway. The trick is to figure out which ones you need to add.

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Regarding how you can lose the temptation of doing math all the time, I'd like to mention the concept of marginal utility. The marginal utility of anything (say, an ounce of water) is how much acquiring one additional unit benefits you. If you're dehydrated, the marginal utility of an ounce of water is very high. As you drink more and more water, its marginal utility diminishes. The same thing happens with mathematical knowledge. Even if you've been consuming new math for years and are anxious to get more of it, there will be a point where its marginal utility will diminish, and you'll be able to focus on other things easily. You just don't know yet what that point is, and maybe you won't reach it even in years, but based on my experience, I expect that you'll reach it at some point.

Answer 19

Something could be useful when Special Force Soldiers are trained, named Devil Training.

It seems the good reasons are related to transcend extreme limit of capacity to quickly extend the comfort-zone and equipped with massively difficult practices in very complicated environments NOT easy ones.

Some parts the same as good mathematical training, one could absorb both broad and important knowledge quickly, and move on quickly, intensively practices to be done in advanced and complicated 'environment' to train both elementary and advanced skills at the same time. Never stay in easy level but just move on even if some skills/knowledge are not absorbed well, since it'll be better as we go further. There's only a fine line between Wisdom and Foolish, wise persons absorb and move on fast with playing around with knowledge like a swimming fish and keep on deepening the understanding; while fool persons only suffer from the pain passively and repeat everything mechanically without thinking.

While some parts are apparently opposite, in Military Training it's obey the orders, no question asked, but in mathematics(natural science) it's break the orders, endlessly ask questions. Plus some passion, curiosity, self-automatic willingness to explore and some luck, it can generate a very beautiful road in mathematical world.

Answer 20

I know there is one fellow here that wants to discourage you from doing your mathematics... He is absolutely right, graduate school is very difficult, post-docs are a nightmare... What is disturbing is that he tries to set you up for a comfortable life... Perhaps this is not what you want. One thing I know for sure is that any human being is "the captain of his soul". If you want to do something, and if you don't give-up you will get there. Nothing worth doing comes easily. This is why there is such a competition for professions such as mathematician or cook. Because the people that want to do it absolutely love it and for them there is nothing better they could be doing. Yes, you will not be paid well, yes some idiot is going to ride a better car than you. The ultimate question however is, if you want to follow your passion to the very final destination where it leads you, perhaps to the abyss, or if you want to never try, and be pushed around by every kinds of people that will tell what they think life is and how they think life should be lived. I know you don't listen to these people anyway because you question even mathematical objects, you ask "why?" which is all you need, coupled with a practical life-approach, for a career in mathematics. I didn't listen either. The same people told me I would get nowhere, that mathematics is too competitive, that there is no room for me there.

On the practical level, it is not normal for mathematics to cause a gain of weight; there could be external factors you should look into. Also mathematics should not cause a loss of social life, perhaps your friends were not the best fit for you?

Concerning academia, inform yourself about the practical means of staying and surviving in academia. You will need a game-plan and a well thought-off strategy always present for the next 3-5 years at every moment in your career until tenure-track. Otherwise you might end up with no job. The most important "currency" in academia are publications, their quantity and their quality. You should incorporate these in your plan very soon. Prestige of the academic institution and of the professors that you've worked with is important in the early beginnings. Look at successful mathematicians (adjust to the level where you want to be in 10 years but always try to do slightly better than they did), their papers, their career progression, etc. to get an idea what is necessary. I wish you the best of luck, no matter what you decide is the correct choice for you.

Answer 21

A friend send a link to this post and I find it very interesting. One thing that is particularly interesting is your conviction that you do not have a talent. How do you know if you have a talent or not? What is talent anyway? Why are you so sure that your peers who get a proof faster or remember it longer have more talent than you?

It is very difficult to explain what mathematical talent is. Most math problems that are worth solving and most theorems that are worth proving take years to solve and proof, so speed or memory will not come so handy while solving or proving these theorems. I believe most humans have enough memory to store all the necessary information in order to work on worthwhile problems for few years (they have enough time to do that). Speed in a three or four year project is hardly ever useful (of course there are notable exceptions but in average I would say this is irrelevant). What actually comes much more handy is diligence, which you seem to have. If you can stay with a problem after few months of failure then you have the right qualities, I think.

One thing people bring up often is imagination. This you can never know if you have or not until you actually start working on problems. You can learn a language quickly but after doing that you may never become a poet. The undergrad math, in fact most math taught in standard classes, is just developing a language which some people learn faster than others and some remember more than others, but what will they do with it isn't something that is taught in those classes.

I would say that mathematical talent is in fact this imagination. To say someone is mathematically gifted is just to say that the person sees more mathematical connections and relations among mathematical objects than most people. A mathematically gifted person has a strong intuition about which line of thought will lead to beautiful new theories and new discoveries. Good memory and speed of performing mathematical computations and logical operations is often mistaken for mathematical talent. It is of course a talent, a very useful one, but I wouldn't call it a mathematical talent.

At any rate, you seem to be far away from a kind of place in your life where you can actually figure out if you are mathematically gifted or not. You can speed up your journey towards getting there by singing up for research projects rather than taking math courses, which are the most deceiving indicators of mathematical talent. You can also take reading courses and sign up for more higher level, like graduate level, courses. There are also summer research programs that you can sign up for.

As for healthy life style, I don't know. I actually think about math when I do my exercise. That is the great thing about the profession but you have to teach things to yourself. For instance, you can teach yourself to think about math when you are doing routine daily things like washing the dishes. But it took me a while to get to this kind of state of mind, as a college student I had an unhealthy life style as well and didn't exercise much thinking its waste of time. But its not! You can teach yourself to think about math while doing it and also it is a way to rejuvenate yourself and deal with the stress.

The social life part is hard. As a research mathematician you do need significant amount of time to yourself. I doubt that there are research mathematicians out there with huge social circles. But with some effort you can have enough people around you and these people, in most cases I know, are usually very interesting, intelligent and motivating people.

As to how to learn math, well, you never know. We all I guess put a lot of emphasize on learning the proofs and the details of it but when we go a higher level we realize that we didn't really understand the proof. So in a sense its actually a pointless activity to really learn the proof of a theorem you have seen first time. Probably doing some kind of circular thing where you learn things and as you go on you come back to earlier things and you re-learn them is better.

But what is more important I think is for you to figure out why you want to learn these proofs so well. There are many more proofs to learn and you will not know all of them by heart ever. In fact what is it that you want to gain out of math? Its best to concentrate on finding what area of math you like and learning that subject with a good professor in a circular fashion where the subject gets more and more sophisticated and important ideas that have been left behind get revisited from time to time.

In short, there is nothing to be afraid of and I am sure you will figure these things out for yourself.