How should I approach taking math tests?

I always do bad on all my math tests yet I do great on projects and homework. Also I like doing research. For exams that have proof based questions I just freeze up under pressure. I just can't do things well in a timed environment. So what are some good strategies for taking math tests?

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Since this question is asking for a list of things, and admits no single right answer, I've converted it to community wiki. –  Zev Chonoles Jul 10 '11 at 15:55
You could try getting more used to timed enviroments. Time yourself doing homework and try to get your time down, at least on the questions you'd normally be confident answering. –  Leonardo Fontoura Jul 10 '11 at 16:09
Get good sleep and nutrition leading up to the test. Warp your mind into believing you are the cleverest person there (even if you're not), and that you will find the trick to each question. When you hit a wall and start to get discouraged, mentally force yourself to "fight hard". If you get stuck and feel your motivation starting to flag, bust out of it by computing special cases, writing down the definitions, or other mindless things that will "hook" your mind on the problem again and provide new ideas. –  Nick Alger Apr 9 '12 at 12:14

This will almost certainly sound like I'm making fun of you, but I assure you I'm not. I give this advice to my students constantly, and for some reason a ton of them ignore it. The ones that take it tend to do extremely well on the proofs I give on tests.

When I have a student say that they panic and blank out so they can't think of any ideas for how to prove something on a test (it sounds similar to your situation), I tell them to just remember one thing: Write down the definitions of the words in the question.

If you are stuck, then try this. Even if you aren't stuck, do this. Of course it depends on the class, teacher, and intended difficulty of the test, but almost any proof you see on an exam in an undergraduate class (that isn't something the teacher intended you to just memorize ahead of time) this trick will get you half way or more to a full proof.

Idea: Converting the words to their definitions will firstly give you some extra words to work with, but secondly it will often give you notation and symbols to start working with as well. Lastly, one of these definitions will be what you are trying to show, so it will give you something to work towards.

I'll give you an example. I taught Linear Algebra last quarter, and I had probably 80% of my students get almost no credit on this question because they refused to write the definitions:

Suppose $A$ is an (mxn) matrix, $v$ is a vector in $\mathbb{R}^m$ such that $A^Tv=0$ and $w$ is in the range space of $A$. Prove that $v$ and $w$ are orthogonal.

This problem caused massive panic because it involved a bunch of concepts and they were being asked to show something original. But look what happens when you calmly write the definitions.

$w$ in range means there is some $y$ such that $w=Ay$. This introduced symbols. Orthogonal means $w^Tv=0$. This gives us something to check.

We're basically done now that we've written the definitions because let's just check if $w^Tv=0$. Well, $w=Ay$, so $w^T=(Ay)^T=y^TA^T$, just plug in $w^Tv=y^T(A^Tv)=y^T0=0$.

See how you can go from having no idea where to start or what to do to an almost complete proof with good direction of what to do merely from writing the definitions.

In all honesty, this trick is good for more than just undergrad courses as well. I used it to great effect on my qualifying exams in grad school as well.

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+1 Excellent answer, Matt! –  amWhy Jul 11 '11 at 0:08
+1. I like the definition trick. And this is also important for understanding in mathematics. –  Jack Jul 13 '11 at 16:20
@Matt $A^T(v) = 0$ means that $v$ is in the kernel of $A^T$, which means that $A_i \cdot v = 0$ where the $A_i's$ are the columns of $A$. This means that $v$ is in the orthogonal complement of the range space. Done. –  user38268 Aug 31 '11 at 7:03

Some basic tips that are often repeated and that work very well :

1. Always begin with your course when revising until you understand every single detail inside and can rewrite yourself the proofs of the theorems involved ( you can also try to find other proofs by yourself ). Write down every single question that pops into your head.
3. If you got satisfied with the answers and fully understand your course, then it's time for training. Pick up some simple application exercises ( your teacher may have given you some, if not, you can easily find them in the internet or by buying a good textbook ), and try to do them until you feel that you can easily apply the theorems you've seen in class.
4. Pick up some problems ( you can find them mostly in textbooks, and if you give us your level and your field(s) we may suggest you some names ), and try to solve them alone. At first they will seem hard to you and you might just look at them with no idea on how to solve them, in that case you might want to ask a friend who's good at maths or your teacher for hints ( of course, you can also ask here, but the disadvantage is that someone here might give you the full answer, which is not a good thing ). As soon as you progress, you will develop an intuition and good strategies to attack problems even if you've never seen them before.
5. If you can get sample math tests that your teacher gave in the previous years, then it would be perfect. You could simulate a real test at home, by trying to finish each one in 2 hours ~ 4 hours ( depends on the nature of the test ) without going to drink, looking who's connected on Facebook etc... => full concentration is needed.
6. Never start to review just one day before the test :-)
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Very good advices. Shouldn't the exercises (in step 3) start in step 1, at least the simple ones? –  Américo Tavares Jul 10 '11 at 16:24
@Américo Tavares : because I don't think it's a good idea if you don't fully understand your course. It reminds me of someone who tried to evaluate the integral $\int x \ln x$ without taking the time to first revise the "Integration by parts" section of his course. –  Bouazza S. Jul 10 '11 at 16:29
I prefer to study theory, then do exercices and go back to the theory, then do more (difficult) exercises, etc. That helps me. –  Américo Tavares Jul 10 '11 at 16:38
@Américo Tavares : yes, it might be good too, only if you can sort out exercises in order to skip those who need notions that you still haven't revised. Finally, of course it depends on each one's way of doing things :-) –  Bouazza S. Jul 10 '11 at 16:56
I find solving exercises helpful when it comes to understanding the theory better. Of course one can't be trying to solve an exercise with no knowledge whatsoever on the subject, but postponing solving exercises to after you know the material perfectly is also not a good idea, in my opinion. –  Pandora Jul 10 '11 at 18:21

My suggestions: When studying, mimic test conditions as closely as possible.

Compose "mock tests" for yourself using questions from old tests, questions from the textbook, etc. Make your "mock tests" look as similar as possible to the real tests in terms of number of questions, number of pages, physical layout of questions on the page, and so forth.

Time yourself taking the mock test. Try to simulate test conditions -- use a similar seat in a similar room with similar lighting.

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This is done by Berkeley grad students, for each other, for the oral exams. Perhaps friends in the same class could work up mock exams and administer to each other. –  Will Jagy Jul 10 '11 at 20:18
In the same line of thinking: try to take "mock texts" at the same time of day that your class meets (e.g., on a day your class doesn't meet, in the same classroom, if possible). "Context of Learning": we are better able to recall our learning, and apply it successfully when we are tested in circumstances which approximate the circumstances in which we learned the material. If you study chewing gum, e.g., have some gum handy when you're taking a test...Lighting was mentioned...perhaps spend more time at your school/college/Uni studying (institutions have certain smells and feels about them. –  amWhy Jul 10 '11 at 22:11
...and since that's where you'll be tested, immerse yourself in an academic environment while studying and taking mock tests. Of course, you'll certainly want to study at home, I'm sure (after all, none of us (err...very few of us) really want to live, 24/7, in a classroom/library/hallway/etc. –  amWhy Jul 10 '11 at 22:14
@amWhy I don't agree with you. This context dependence only occur when one studies in anninvariant environment. I've read about some research showing that studying in a wide variety of settings helps you learn better overall, and be able to recall in any situation. –  ErikE Jul 11 '11 at 7:28

Your post has moved me to tears,Terry. It's reminded me of my own problems and as someone who's lived through this, I feel compelled to speak.I hope the others will forgive me for waxing personal on this,but it's a very painful issue for me.It may all be irrelevant as you may not have this problem as I did. But I wish someone had shared this with me at the beginning of my career.

I was also an extremely talented undergraduate-I was once an honors student in a double major of mathematics and biochemistry. I had the chops for graduate school at Harvard or Yale if I could have gotten my act together. My career was hampered and my grades very negatively affected by having the same problem with exams,Terry. Unfortunately,it was greatly amplified by being at the center of my father's decade long battle with cancer. I hope you don't take offense-but it's important to know whether or not this is just butterflies or a real underlying condition. If the former,then there's some good advice above and I'm sure you'll get more on here.Practicing test conditions is a particularly useful strategy.

But if it's the latter,I STRONGLY suggest you seek medical consultation.It may end your career otherwise.Exam performance is the single most critical determinant of whether or not you will advance in the academic world or become the joke of the department that the chairman points out to the new honor students as a cautionary tale.

Trust me. I was also someone who was better then nearly everyone else-until test time.I made it a graduate student-but it wasn't in anywhere near a top program and my less-then-stellar grades will haunt my career until I make a substantially significant publication. Which means I may never make it as mathematician of any distinction.

I swore I'd do all I could to make sure any other talented student with this problem doesn't make the same mistake I did,ignoring it and hoping it goes away. It doesn't. Don't make the same mistakes I did. It may be more serious then simple butterflies-in which case,none of the techniques suggested here will have much effect. They may have methods that may be able to help you. I don't know if you have access to such medical treatments,but if you do,get yourself checked out.

I hope it's not. I really do. Good luck.

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Addendum: I want to reinterate that I hope the members of the Exchange-many of whom also are regulars at the professional-level board Math Overflow-will forgive my waxing personal here.I also wanted to add Matt gave a particularly good suggestion-it's something I do regularly on solving problems and proving theorems.It's a lot trickier to do in a test condition,but it's doable if you get a lot of experience and increase your speed enough.It may not get you the answer,but it'll certainly point you in the right direction! –  Mathemagician1234 Jul 11 '11 at 17:56