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

Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exercise, I've gained $30$ kg, I've spent countless hours studying at night, constantly had sleep deprivation, lost my social life, and developed 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. Before, I've been accused of being a formalist but I don't consider myself one at all. However, I admit that I am a perfectionist. 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 but, for some reason, I can't. I don't understand things unless I understand them (almost) completely. They learnt "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, all are $10$s. He mentioned 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. Did you manage to study everything on time, AND sufficiently rigorous, that you were able to understand it?**

ADDITIONS:

Frequently, I tend to be the only one who find serious issues in the proofs, the formulations of theorems, and the worked out exercises at classes. Either everyone else understands everything, most or doesn't understand and doesn't care the 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 mentioned I'm very precise. How is this precise, when the theorem doesn't hold in its 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. Furthermore, 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, one to actually walk + one 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. To add on, I have a bad habit of having difficulty starting things. However when I do start, I start 'my engine', and 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 had more math than I can shove down my throat which I have, for mental preparations to begin studying. But, as my engine really starts 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 something useful before I go to sleep. I know this isn't supposed to happen if one loves mathematics. However, 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. Mathematics stops being enjoyment/fun and becomes hard work that just needs to be done.

Solution 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 each 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 you 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".

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.

Solution 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.)

Disclaimer (Dec. 25, 2013): This answer was written when I was 16 years old and does not necessarily represent my current views of mathematics. (Some points, e.g., "write down all of the mathematics you learn" is not something I would recommend to anyone today.) But I leave my answer here because I think it is overall reasonable advice and has clearly been useful to many people as is evidenced by the 77 upvotes.

Solution 3:

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 principal 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, 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.

Solution 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.

Solution 5:

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.