How to effectively study math? [closed]

This is a hard question to answer, because the answer would depend a lot on personal aspects of your situation, but there are some general points of advice one can give:

(1) Try different books. It may be that the particular textbook you are using doesn't click with you, but for classes like algebra and other pre-calc courses, and calculus itself, there are hundreds of texts available, and some may fit with you better than others. If you go to your local Borders or Barnes and Nobles, there will be a shelf of math books devoted to these topics, and you could look through some of them and see if they suit you. Also, your college library will have lots of books like this too, which you can browse through.

One thing to remember is that different books might be good at different things: your textbook probably has lots of exercises, and you will be able to find other books with lots of exercises. But perhaps you can find different books which don't necessarily have as many exercises, but have better explanations. So you can try to combine different books: some you read for their explanations, others you use for their exercises.

If you do use different books besides the one in your class, remember that sometimes the notation may be a little different, although in precalc and calc books, most terminology is standard: sin, cos, tan, conic section, polynomial, etc. will all mean the same thing. (But some books will use notation like $\sin^{-1}$ for inverse trig functions, and others will use arcsin, etc., instead.)

(2) Practice basic algebra: It is very common, when students make mistakes in precalc or calc classes, that the source of the mistake is weakness at algebra. Practicing algebra will help with everything else that you have to do in math. The skills will be directly useful, and lots of other manipulations you will have to do in more advanced classes will also be similar to the skills you build up by practicing algebra.

(3) Practice with numbers: If you don't think about numbers much in general, you will have trouble with other things in math, because you won't be able to relate them to concrete things. E.g. when you plot a graph like $y = x^2$, you want to be able to easily realize that a point like $(12,144)$ is on the graph because $12^2 = 144$, while $(11,120)$ is not on the graph (because $11^2 = 121 \neq 120$), but is pretty close to the graph (because $121$ is not all that far from $120$). Day-to-day life gives chances to practice arithmetic; try to take advantage of them.

(4) Try to learn from your mistakes: One advantage of mathematics is that when you make a mistake, there will be a specific reason as to why; i.e. there will be some particular thing you did wrong. Try to find out what it is in each case, and resolve not to make that mistake again.

One aspect of this is that math should make sense. If it's not making sense to you, i.e. if you can't work out specifically what you are doing wrong in a given situation, try asking your professor or tutor again.

One thing that can happen, because of the cumulative nature of mathematics, is that several confusions can become combined in an answer, and then it can be hard to figure out what particular thing went wrong. In situations like this, do your best to break the computation down into small steps, so that you can identify what you did right or wrong in each step separately.

(5) Write down all your working: Use a lot of paper, write down all your steps, make them clear so that anyone else (or you in a few weeks time!) could go back and read them and understand what is going on. If you lay out the whole chain of your computations clearly, it will be easy to identify weak links later. If you skip steps, the whole thing is more confusing and it's much harder to learn anything from it later.

(6) Try to recognize when you understand something and when you don't: Probably arithmetic makes sense to you. When you learn another piece of math completely, it should make as much sense as arithmetic does. E.g. a common mistake in algebra is to write $(x+y)^2 = x^2 + y^2$, but to someone who is good at algebra, this looks just as wrong as $1 + 1 = 3$ does. If it doesn't look that wrong to you, it means that you have more work to do in building up your algebra skills.

I wish I had better advice on how to do this. One thing you could try (say for the wrong example with squares above) is to plug in some random values of $x$ and $y$ on each side and check that for most of them the alleged equation won't actually be true; this tells you that the equation between $x$ and $y$ is wrong. (If it were right, it would hold for any value of $x$ and $y$ that you plug in.) I don't know how helpful this will be though. One thing I can say is that people do often check their algebraic manipulations in this way: after making a complicated manipulation in some algebraic equation, they may plug in a few random values just to make sure that the equation is still correct. Also, most people who are good at algebra go back and plug in their solutions after they have solved an equation, just to make sure that they really did solve it correctly. So this is a good habit to get into (and practicing it will also help you practice your arithmetic).

As I wrote at the start, I don't know how useful this advice will be to you in particular; it is based on my general observations of students after many years of thinking about and teaching math.

This advice from a teacher I had, long, long ago has stuck with me all along.

"There are three ways to react to any [troubling] situation:

1. You can get frustrated... ARGH!!!

2. You can feel intimidated... :-(

   or...

3. You can be INSPIRED!

In other words, try to catch yourself when you're feeling overwhelmed or frustrated; if you can reframe the situation as a challenge, rather than a threat, you're more likely to feel inspired, and that, in turn, will give you the stamina to work through what's confusing and get you to the other side. Persistence is key.

And also, remember the progress you've already made...There was a time when what is clear and obvious to you now was confusing and beyond your grasp, no?

I should also add: we all have run up against brick walls, here and there, along the way; being confused, stuck, overwhelmed, etc..., while learning mathematics is not an indication that you're "stupid" or "not cut out for math"...I think one of the most important factors determining success or failure is how one responds to feeling stuck or totally confused, which in turn influences how persistent one is. Persistence is also one of the most important factors determining success or failure...This applies to learning in general, not just learning mathematics!!

I've answered this question several times-once on Math Overflow and once here. I don't really have much to add to that answer because I think it's great advice (not to break my arm patting myself on the back about it) because it was arrived at through a lot of trial and error, pain and suffering.

Firstly,although I firmly believe mathematics has to be learned actively and looking up answers should be something you try to avoid, there has to be a practical limit. Most of us have gotten a stubborn streak with a particular homework problem as a student where we literally waste days trying to solve it-we're not gonna let it "beat" us. Part of it is just stubbornness, but beneath it is a deeper fear that our inability to solve a problem with no help is the dreaded "wall" that shows we ain't as good as we think we are and it's the first step towards ending up mopping floors outside our more brilliant classmate's office at Princeton. This is a lie, of course-with the exception of the truly gifted, all mathematics student struggle with proofs and computations.

More importantly,since we live in a Real World where there are deadlines on assignments and time limits on exams, such thinking will be very self destructive if it's not controlled. Finding oneself when time runs out on an exam having spent all the time on a single problem and getting a grade of 7 out of 100 for such stupidity is not a good day.

Personally,I think all textbooks regardless of level should come with complete solutions manuals. I know,I get a lot of flack for that,but I think having access to the solutions is a very good thing for students to have because it allows them to set a limit as to how long they'll work on a problem by themselves without solving it. "Oh,but then they'll just look up the answers and get an A." That's a facecsious argument to me because even if their professor is irresponsible enough to grade them solely on work they can look up, sooner or later,they will be required to find answers to even more difficult problems without access to solutions.

A middle ground solution to the corundum is to have textbooks with good, detailed hints . In my experience, a good, well-worded hint is usually good enough for a hard working mathematics student to point him or her in the right direction,they usually don't need more then that to get unstuck. But I'm getting off topic here.

My point is that although certainly you should make every possible effort to try and work things out yourself, there comes a point where it becomes self defeating and you have to either look up the solution or ask for guidance. How long you're willing to work before considering asking for help is a decision you have to make for yourself, but after being a student for some time,it's not hard to work out a reasonable boundary to set for yourself on this.

I'd like to close by telling you how I study. I generally make up a large, detailed batch of study cards-2 kinds for mathematics; one containing theorems and sample exercises and the other definitions and/or examples.The definition and example cards are the critical ones for studying. This is what furnishes the basis for understanding mathematics. For example, to understand the Cayley group isomorphism theorem, you have to understand what it means to have a permutation on a group. You can then try and re-express the result in terms of other concepts. For example, you can think of the Cayley theorem as stating there is a fundamental group action of every group on itself. However you do it-absorbing the definitions and what they mean is absolutely critical. Test yourself numerous times to see if you've absorbed and understand them.

For the theorem cards, this is where things get creative. Study them with a pen and paper in hand.I generally write the statement of the theorem on one side and the proof on the other. Do not look at the proof-try and determine the proof yourself directly from the definition. Wrestle with it as long as you possibly can before turning it over. Then if you can't reproduce it-put it on the bottom of the deck and move on to the next one. Do this until you have 2 separate piles: the ones you can prove and the ones you can't. Then do it again until you can prove it. This works,trust me.

Practice a lot. Get lots of paper and pencils. Post it notes, index cards, and folders. Get organized and make sure you get enough sleep. Math can be strenuous if you are tired. Take breaks. Borrow books from the library at your school and get together with your professor during office hours. This might help. =)

Here’s the link to a website you might want to consider:

http://www.mathreference.com/main.html

It “is essentially a self-paced tutorial/archive, written in English/html, that takes the reader through modern mathematics using modern techniques.”