How to study math

I would go to the library and rent a study carrel and then just stay there for the time I'd allotted, even if that meant I just sat in the chair and read the graffiti left by former students. That time was studying time and I'd stay there whether I studied or not. No laptop, no phone, just me and a big stack of blank paper, pencils and books. I'd almost always get something done. If the library was closed I'd just sit in an empty classroom. Funny thing is, I'd see a lot of other math majors there, too. Especially the ones I thought were just naturally awesome at math. Nope, at the library all the damn time.

Work problems. Work problems. Work problems. Work problems. Work problems. Work problems. Reading is not studying. Discussing is not studying. Memorizing is not studying. They are all recreational activities that make your life interesting, but they won't help you on tests. Working problems *is* studying. Not very exciting, I'm afraid. But, it took me an embarrassingly long time to learn it. The great thing about solving problems is that you can assign yourself a certain number every week, and it's clear when you've done them "enough." Unlike reading, where one can always do more or less and feel equally fulfilled. Get exams from previous classes. If there's a department club on campus, see if they have a file drawer of past exams. Otherwise, work textbook problems. "All the numbers divisible by 4 in one chaper per week" is the kind of thing we procrastinators can commit to. (Also, for me, "never work with other people" was an important rule to learn. But, most people I know who've done well in academic STEM fields disagree with that, so I don't recommend it.)

You just gotta write write write write proofs. I took a real analysis class where 100% of the classroom time was students proving stuff on the chalkboard. Every theorem, easy or hard. It was the best class I took in college by far. http://www.jiblm.org/downloads/jiblmjournal/V090212S/V090212S.pdf was the course material we used. If your class is more lecture-heavy, try to work together with other students in the class. Writing proofs is a lot easier if you have another person to bounce off when you get stuck. You will be forced to study earlier than the day before because of scheduling compatibility.

Which class(es)? It's probably a case of doing as many problems as you possibly can. Also, how many classes are you taking? I've heard recommendations of 2-3 hours of work out of class for each hour in class. I think it averaged out to be close to the in the end, but some classes took much more and some took much less. Do you have a good feeling for how much time you spend now and how much you'd like to spend? It sounds like there are two issues here: 1) scheduling the studying 2) how to study. Scheduling the studying Setting aside X amount of time rather than saying I'd work until I finished Y problems helped. That I way I was sure to put the time in and wasn't discouraged if I didn't get too far on one day or another. Proofs can take a fair amount of sitting and staring time. Scheduling study time like class time or any other appointment was helpful. If I put it my planner that I'd study at between X and Y hours at Z location, then I was much more likely to do it. I also knew that I'd have my study time covered so I could go do non-school things too. I roped friends into this too; they knew I was unavailable during set times and happy to hang out others. Keeping the blocks of time relatively small (no more than 2 hours at a time in one block) helped ensure that I'd stick to it regularly. I did do larger blocks too, as needed, but smaller ones were easier to manage. Physically moving to a study location made a big difference for me. I found that studying in a location where other people were quietly studying (often using my headphones to block out any small noises) was a big help. I'd be less likely to goof off or get distracted. This can be the library, but it can be other locations too. I used the quieter CS computer labs often. Once at the scheduled location, I found a http://en.wikipedia.org/wiki/Pomodoro_Technique to be helpful. Work for 20-25 minutes, take a 5 minute break, and repeat as necessary. You may want to put at least one session down on your schedule each day. 30 minutes is usually pretty easy to fit in somewhere each day. Suggestions on how to study: I found study groups helpful, but needed to have both study group time and study alone time. Scheduling both helped. Oh, and scheduling time in Office Hours can help too! For memorizing material (to make working problems easier/faster as there can be a fair number of definitions to unpack), I found making flashcards helped. I also took some studying to a decorating level: I got a large pack of multicolored post-its and used each one as a flashcard, then stuck them to my closet doors. That way I studied once by making them and refreshed my memory by looking at them every day. They looked nice and were an interesting conversation piece. You could set aside 15 minutes or so for doing this each morning, either creating cards for new definitions or reviewing the ones you have. When working problems, once you've found a solution, try to look at how you could solve the problem differently (more efficiently?), how it relates to previously solved problems, and how altering the problem would change the solution. (See Polya's "How to Solve It") I didn't do much of this until it was part of a course I took, but it really helped with understanding. Using a whiteboard (or sometimes a chalkboard) helped me write out my thinking when doing problems. There was something about the space and easy erasure that helped with my thought process. In an odd way, using LaTeX helped me work problems better too. I'd typeset the problems on the problem sets the day I got them, so I'd have read them at least once through so they could be churning in my head. Typesetting my solutions helped focus what I wanted to say as well. Good luck and feel free to MeMail me if you want to discuss any of this further.

Most of what you're talking about is just study habits, which seems like simple stuff to some people, but is actually http://www.nytimes.com/2010/09/07/health/views/07mind.html, so you're probably going to have to survey the landscape, get some ideas, and then try to track what seems to work for you. Here's a few ideas I think are worth considering, though: * Study groups. There were some classes where the materials were good enough or the math came easily enough I could get by on my own, or at least fake it, but there were also classes where there was just no way I was going to be able to do that. Study groups provide a socially reinforced time to dedicate to the topic, they provide peers to be accountable to have tried at least something when you show up, they provide people you can explain things to, and they sometimes provide people who can help you through a concept you're not getting. They can also provide you a sense of how well you're doing in the class. I withdrew (both officially and unofficially) from a few classes because I had no idea that my apparently terrible scores were actually at least average on the curve (sometimes in-class dialogue will tell you this, but other times, you only hear from the handful of people who are doing well with the material and everybody else just sits in confused silence or tries to copy down notes to try to make sense of later). If for some reason nobody wants to do study groups, hitting a prof's office hours semi-regularly after doing your homework sets can be a decent fallback, depending on the prof. Doing this anyway isn't a half-bad idea. * Try to keep your head as clear as possible going into other studying. If there's other mundane but consequential stuff you need to take care of, schedule time to take care of that earlier in the day. If there's anything you can't do before you study, make sure you're writing that stuff down somewhere and don't need any headspace to remember it. And I know this is college we're talking about, but sleep, eat sensibly, and get modest exercise so your brain is working well when you tackle the topic at hand. * Talking of which, some people find they can actually work proofs/problems in their head while walking or lightly exercising -- one prof told me he used to walk a few hours to UW Madison and used that time to work. This wasn't reliable for me, but it was sometimes helpful after I'd spent some time plugging away in front of a notebook first, and it's a good two-birds-with-one-stone. * You might try http://en.wikipedia.org/wiki/Spaced_repetition. I wasn't aware of this when I went through my program and so I can't attest to its usefulness, but seems like it could help. * Meta-manage as well as you can. Get tips for which classes to take at the same time (overlapping material really helped me a few times) and which classes/profs to leave lots of time for. "never work with other people" was an important rule to learn. But, most people I know who've done well in academic STEM fields disagree with that, I would also disagree with that -- I really wish I'd engaged with my classmates and profs more, and I find that outside the academy I miss the reinforcement that can come with the community. My own opinion is that the important thing is to (a) survey the material solo before you do group study to get yourself conversant with the basics and potential tricky parts and (b) at some point afterward verify to yourself that you can work all the way through without outside assistance. I would conjecture that "Never work with other people" is an overly specific case of the more general guideline. ;)

There are lots of good suggestions above that specifically address grades, good study habits, and time management. I'll add a suggestion on how to increase depth and retention of understanding. Study groups can be good, but I've found one of the best ways to simultaneously identify gaps in my understanding and deepen it is one-on-one discussion with someone with a level of skill and enthusiasm for the material similar to yourself. Try to identify someone in your classes who fits that description and get together to discuss whatever you mutually find interesting about the material: a particularly difficult or subtle proof, visual intuition for some concept, the professor's pedagogical approach to a certain topic, cross-class/subject connections, etc. If you can find the right partner and the right dynamic, such conversations can give you both a steady stream of those "hmm, I never thought about xxx like that" moments. Some caveats: 1. If one of you is getting plenty of "a-ha!" moments but the other is not, you are probably not a good match. 2. If you can find someone in multiple or all your classes, all the better. 3. If you can find someone who can do this across multiple semesters, even better! 4. Be careful the conversation does not just devolve into talking about how "cool" X or Y is. Your explicit goal should be to push each other to that edge of knowledge where you feel some mild discomfort/anxiety at the lack of understanding and then help each other map out the new territory and connect it back to your prior knowledge. 5. In my experience, this really works best with the intimacy of just 2 people, possibly a third if the conversational chemistry is just right. Once you hit 4 people, the dynamic changes too much and at least one person probably isn't getting much or contributing much. 6. Even just a couple hours a week doing this can significantly deepen your understanding. Good luck!

What classes? I made flash cards with equations for things like Stokes’ theorem written on them when I took Calc IV back in the day. Linear algebra, I just made sure that I could do everything by hand. One thing that helped was teaching the material to classmates as a study method. [I finished a BA in math a few years ago]

Start your problem sets the day they're assigned. Look them over and figure out which problems you have an idea about right away and which ones look harder. Polish off the easy ones in your first study session. Start thinking about at least one of the harder ones. Parcel out 3-4 chunks of time over the course of the week to finish the rest of the problems. The nice thing about this is even if you don't solve a problem, it'll percolate in the back of your mind and suddenly when you're coming back from lunch you'll know how to do it. Another thing to do as you go along is to try and explain why the result is true in 1-3 sentences stripped of most mathematical jargon. Then try and see how this idea is reflected in the proof. So for something like the Intermediate Value Theorem, it's true because if you don't pick up your pencil, then you have to hit every value between your starting and ending value. And then you look the proof and you see where you're really using continuity. Especially in Real Analysis, it's easy to get tangled in all the epsilons and deltas. The key is to look at what's going on beyond all that.

Step one is do all the homework, which means not starting it the night before. In my experience, there was no time to do other problems. I seriously don't remember the last time I finished a homework set early. Upper division linear algebra, probably. In other words, my second semester of college. TeXing your homework is a good suggestion, if it's not required. It also means you don't lose your old homeworks. I'd write up partial solutions partway through the week and print it out with blank space to fill in the rest. Until grad school, I studied for exams starting roughly a week before and writing out all the theorems and definitions by hand. And then reading over the homework. In grad school, I added flashcards and then basically switched to flashcards rather than writing everything out. You want definitions, named theorems (or any theorems you can make up names for), examples that distinguish property A from property B, etc. Then you want to know them cold, verbatim. Basically, I'm pretty crap at taking tests and the more I could do without thinking, the better. (Doubly so for professors who ask definitions.) In grad school, I also switched to trying to redo as much of the homework as possible (precisely why it took me so long, I don't know--I got burned by a homework question that appeared on that upper division linear algebra final where I couldn't quite remember how it went). I'm still a fan of writing things out though, particularly if there's a 'story' of some kind. The q-binomial theorem was crap as a flashcard, so I ended up writing out the whole tale repeatedly until I could do it smoothly. But then I was also anticipating being asked to explain it.