How to study math

March 5, 2013 7:44 PM

What is the best way to manage time while studying for upper-division undergraduate math classes that are proof-based/STEM classes in general?

I entered college as a social science major and I ended up a pure mathematics major with a minor in computer science. I'm taking my first proof-based upper-division courses this semester and I'm having trouble staying afloat.

I have the bad habit of cramming. I will often neglect other classes and just study for one class for every free hour of the days leading up to an exam, then repeat the process all over again next week for another class. This has worked so far, but I just recently realized what a waste it is. I have absolutely no free time to do anything other than cram and do homework.

I spent all weekend studying for my real analysis class and I really started to understand, and even enjoy, the material. Going through my notes, I can't believe how amazing some of the stuff we've learned so far is. I also can't believe it took me so long to realize that my cramming is seriously detrimental, not only to my grades and knowledge, but to my enjoyment of the material. Now that the classes are challenging and interesting, I really want to learn instead of just do well on tests. It is such a waste of time and money to cram for exams rather than actually learn.

My only hangup is I have absolutely no idea how to transition from cramming/memorizing for tests to studying the correct way. My mindset has always been study for Monday classes on Sunday, Tuesday classes on Monday, etc.. I want to spend a set amount of time each day studying for each class. I also want to set aside time for hobbies, reading or studying other material. My spring break starts next Friday and I plan on catching up completely so I can finish the semester using this studying method.

I do not necessarily need advice on HOW to study for these types of classes (cross referencing proofs with other textbooks, doing extra problems and asking for clarification from profs/TAs works for me). I just need advice on how to implement the "study a little every day" method I always see suggested. Has anyone actually used this studying method and has it actually worked? Does anyone have any tips on how to manage my time to fit an hour for each class every day?

I would also specifically like to hear from math/STEM majors about your study schedule and study methods (be as specific as you want). I would appreciate hearing how someone in the same types of classes goes about studying. Thanks for any advice!

posted by Hey Judas! to education (15 answers total)

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]
posted by oceanjesse at 7:50 PM on March 5

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.
posted by ZeroDivides at 8:00 PM on March 5 [1 favorite]

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.)
posted by eotvos at 8:14 PM on March 5 [1 favorite]

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. This 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.
posted by scose at 8:19 PM on March 5 [1 favorite]

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.
posted by matildatakesovertheworld at 8:41 PM on March 5

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 Pomodoro-style 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.
posted by wiskunde at 9:00 PM on March 5 [1 favorite]

Most of what you're talking about is just study habits, which seems like simple stuff to some people, but is actually debated, 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 spaced repetition software. 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. ;)
posted by weston at 9:14 PM on March 5 [1 favorite]

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.
posted by hoyland at 9:16 PM on March 5

(a) survey the material solo before you do group study to get yourself conversant with the basics and potential tricky parts and

When you do homework with other people, always show up having tried a substantial portion of the problems.
posted by hoyland at 9:18 PM on March 5 [3 favorites]

Here's some techniques that work for me in studying for proof based math classes, similar to what has been suggested so far.

-After first reading a proof or reviewing notes from class I didn't quite take in, I look for where the assumptions (e.g. if a group is commutative, a subset is convex, etc) are used. This can be pretty obvious or pretty subtle. (You can also go further and try to think of counterexamples where the proposition would fail if the proposition would not hold.)

-When I'm going through a textbook that I've seen everything from, and encounter a theorem, I try to prove it without reading the proof first. If I can't do that, I let myself read the first sentence or so of the proof and then try to solve it, and so on. Particularly useful if your professor is the type to ask these book proofs on an exam.

-2nding Wiskunde's suggestion to TeX up problems early. When I'm feeling lazy, I tell myself that I just have to TeX up the problems from my classes, which is a pretty mindless activity. Sometimes I then have the motivation to solve them, sometimes I don't, but either way, I'm aware of the problems earlier, as well as the difficulty of the problem set. This motivates me to put aside the right amount of time to finish it. (TeXing problem sets in general helps me. They look more serious to me, which helps me feel better about my classes.)

All of these things are great because they force you to actually take time to do them. Sometimes my problem doing math is that I can read a dense chapter in 20 minutes or look at my scribbled notes and think I'm ready for a test, whereas I actually need to go and do problems; similarly, I can put off problem sets forever because I don't actually know the problems I'm going to do. The techniques above stop me from doing these sort of things.
posted by precession at 9:22 PM on March 5

I just need advice on how to implement the "study a little every day" method I always see suggested. Has anyone actually used this studying method and has it actually worked? Does anyone have any tips on how to manage my time to fit an hour for each class every day?

I'd suggest starting with a study skills book, which will talk about how to set up a schedule. The book I used was Hawes Guide to Successful Study Skills. It recommended dividing up your time outside class into one-hour blocks, and allocating the blocks to each class. You might have 9-10 am Monday, Wednesday, and Friday reserved for your real analysis class, for example. An example template.

When I set up my schedule for each semester, I followed the "three hours of study for each hour in class" guideline. I allocated an hour before each class for preparation -- I would usually skim the upcoming chapter in the textbook (making it easier to absorb the material during the lecture), and spend some time reviewing previous material. It wouldn't necessarily be the hour just before the class, as long as it was on the same day.

Similarly, I'd reserve a block after each class to review the material, making it easier to retain. And I'd reserve a third block to work through the problem sets.

Around exam time, of course, I'd have to adjust the schedule to allocate more time to studying for key exams.

I found this kind of weekly schedule worked really well for me. (I was doing a combined honours degree in math and computer science, and I compressed it so that it would fit into three years instead of four, so I had a fairly heavy load.) When you get into the routine of following a fixed schedule, you don't need to spend time deciding what you're going to do when you sit down to study, you don't need to exercise as much willpower to get down to work, and you don't need to worry about making sure that you've allocated enough time to each class -- you just have to follow the schedule that you've already worked out.
posted by russilwvong at 11:09 PM on March 5

Nthing doing homework when it is assigned. I just started taking proof-based math this quarter, and doing the homework the day it is assigned has changed literally everything. The first third of the course I did the homework spaced out a few days before it was due (like, it was due on Wednesday, so I did some on Sunday, Monday, and Tuesday). I ended up being confused in class and not keeping up with the proofs, because I hadn't practiced what I'd been taught. For the second third, I started doing homework the day it was assigned, even when that day is a Friday. The change has been dramatic. I'm actually able to keep up with my Austrian instructor, and the other day some guy came up to me and was like, "hey, you look like you are someone who's been on top of this shit...*"

If you do the homework when it is assigned, you'll naturally study a little bit every day, and naturally enforce what you've learned. And you won't have to do ten hours of homework the night before it's due! This really only works though if the class is an hour long and meets several days a week. My math class meets MWF, so I end up doing math frequently, but I am in an econometrics class that meets for two hours, two days a week and it has been awful. Math classes aren't meant to be taught in huge chunks like that!

For how to study, google "Math XXX [school name]." You'll find that professors don't always delete their websites, they just remove them from the school page. Then, do the midterms they've posted on there. Also, you can naturally use the book to space things out to review before tests (sections 3.1, 3.2, 3.3 on one day, 3.4, 3.5 on another, and so forth).

You asked if anyone has implemented this strategy. I think you are confusing "study a little every day" with "do math every day." In some classes, this will be "studying," because there won't be any graded homework**, but in other classes, you will do the homework. What this means is that you shouldn't go to two classes in a row without doing some homework in the middle to reinforce what you've learned. And yes, it is very, very successful :).

*I'm really not. I hate this math class.
**Whenever a professor doesn't have homework it makes me want to cry.
posted by obviousresistance at 11:33 PM on March 5

Since you're taking Real Analysis, I'd suggest reviewing a list of counterexamples for each topic your course is covering. By reviewing, I mean reading through and working out for yourself why each is a counterexample. I've heard good things about Counterexamples in Analysis but your professor may be able to recommend another book. Knowing both examples and counterexamples helps cement understanding.
posted by wiskunde at 6:45 AM on March 6

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!
posted by stoffer at 7:21 AM on March 6 [1 favorite]

You've gotten great suggestions so far. When I was in school (chem major, math minor), I studied from after breakfast until my first class every weekday. That alone was huge. You're not doing anything that early anyway, get a couple hours of studying in. I also had a study group for each STEM class. We tended to meet either right after class, during the prof's office hours, or the night before an assignment was due or before an exam. The key to making study groups work well is preparation. I would never meet with my number theory group without having gone over the material and attempting to work the problems first and I expected the same from my partners. Doing preparation like that helps you identify the things that you need help with and puts you in a great position to be helpful to others. I've found that explaining a concept to someone else really helps cement it in my own brain.
posted by tealcake at 8:26 AM on March 6