I'm currently an undergrad math student and I have been wondering if being able to perform well under time constraints requires a deeper knowledge of the material than performing under no time constraints, specifically in the context of test taking. My instincts tell me that even though I may be able to do well in a relaxed environment not doing well under test conditions probably implies that I do not know the material as well as I should. Any thoughts?
It is all about confidence I think. How much confidence you have in your ability and comprehension. How much do you think you understand the material and how well will you do under time constraints and pressure.
Now confidence I think comes from a bunch of different things. One is your natural disposition. If you are very confident in general, good. If you naturally have problems taking tests/under pressure/with time constraints, not so good. But things are not so dismal...you can definitely improve things. Its just that you might find better general advice about that particular aspect somewhere else, not so much on a math forum here.
Besides, natural disposition, there are two more things I think which we can talk about here.
1.One is how much time you put into study and how much time you spent on math, attending class, paying attention to lectures, taking the right amount of notes, going over the notes, studying proofs/techniques, trying to reproduce them, doing homework, and studying discussing with others. Taking the right amount of notes is important because too much then you spend all of your time just literally transcribing the lecture and you don't have time to actually listen and understand what the teacher is doing. Too little, then you will forget some detail or the other and won't be able to reproduce/remember it later. Taking good notes is a very difficult skill to master but very useful once conquered. To this day I actually still have my original handwritten notes from my undergrad days which I occasionally consult.
Some of this is out of control for example the quality of the teaching. If the teacher sucks, then notes/listening/showing up to class won't help at all. Others are in your control for example if the textbook sucks, there are plenty of others you can use as supplement and the internet has made help from others even easier (like using this forum here...please don't abuse it though :-) for example to get us to do your HW for you without you even trying). Doing HWs together in group, sharing notes, asking question to your peers (with the right people) and answering their questions if you guys don't understand something works wonders. Because there's a level of disconnect between your professor and you but your classmates think exactly like you at your level and they will explain it in terms you understand because the day before they probably made the same mistake. Lastly to have faith in yourself, you need to ask how much time did I spend on this. Did you spend enough time...whatever a fair amount is? Remember math isn't a spectator sport.
2.The second important thing is the context of the test. Is it a quiz for a class, test for a class, a final for a class, a friendly competition, a competition with a significant prize at the end, a standardized test like the SAT/GRE, an entrance exam to a school/department, Ph.D level preliminary/comprehensive exams, or what. Is it in high school, lower level undergrad, upper level undergrad, or graduate level? How important is it and how much do you have riding on it? Will you be graded on an absolute scale without comparison to other test takers (like how quizzes/midterms/finals for a class can be) or will you be graded in relative to others performance (like SAT/GRE and doctoral comprehensive exams, on a curve like most grad classes)? Is the scale just pass/fail (like quizzes for a class or doctoral comprehensive exam) or is it much finer (like a percentage or percentile, for SAT/GRE and class midterms and finals)? Are previous tests available for study? Are solutions from the past available? The difficulty also matters of course. All of these things need to be taken into account. Then develop a strategy for the study. Streamline it and then execute it.
This missive can get quite long explaining everything here of course. If you have a particular situation, feel free to tell us and we'll help you out as much as we can. I have done this for a decade going through ALL of these different situations multiple times getting my Ph.D in math. I am sure there's an abundance of useful advice available here from others like me if you want to be specific.
IMHO performing under time constraints has very little to do with depth of understanding. Amount of practice and good "exam technique" are very important, perhaps more important than depth of understanding for doing exams.
i suppose it also depends on the material you are refering to. Let me clarify with an example.
For a calculus course you would need to be able to use different kind of integration methods.
At first you will solve exercice in which they tell you which technique to use. This is to train to technique. However you cannot stop there. You then have to do mixed exercices and then exercices that ask for combination of these technique. One you are a point you can "predict in a sense" where your integral is going and if that takes you closer to actually being able to solve it you mastered this skill. Then there is the speed part. Being able to do some integrals at home is not the same as doing difficult exercices on a test. the actual manual manipulations should go very easy.
Typically I find that on both these points students think they mastered the skill before they truelly mastered it. When the question on test get more complicated this shows in the amount of time needed.