How should I approach taking math tests?

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.

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.
2. Ask your teacher, classmates or simply this site to answer your questions.
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 :-)

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.