How to minimize the chance of making an error in algebraic calculations? This is a question about solving problems under significant time constraints, for example under exam conditions. Even though this question might be seen as "subjective", it is seriously related to many people who study mathematics, so an answer will be very helpful.
The question is simple: what are some ways to reduce the amount of errors in algebraic calculations, for example integrals, derivatives, series expansions, and things like that?
This is important in pure maths, when we are, for example,  doing proofs in analysis/number theory. Needless to say, this is very very important for applied maths and physics.
Apparently, if one is really concentrated on finding the method to solve a problem, he/she becomes much more likely to make small mistakes along the way. Of course, such errors are not an issue if one has plenty of time to check, but it does become an issue when time is limited.
 A: This is arguably more a question for psychologists than for mathematicians, but here are a few techniques which could be helpful:

*

*Check your work! If you have more than one approach which will generate a valid solution and have time to do so, test both. The odds that you'd make the same error in both cases are rather slim. Better yet, if you have a way to verify a solution which does not involve the same degree of effort as solving it in the first place, use it! (i.e. if you are integrating, take the derivative, or if you are finding eigenvalues and eigenvectors of a matrix, actually multiply each of them to be sure you have the correct result).

*Verify calculations, signs, etc. at each step rather than waiting until the end to (maybe) spot an error. This may double the amount of time spent crunching numbers, but it reduces to near-zero the amount of time spent correcting past computational errors and doing work which you'll later have to throw away.

*Verify whichever computations you can using an independent source. Obviously this is not suitable for an exam under most academic integrity policies, nor would it be ethical to use an external source on homework to do anything more than check for correctness (even that may be questionable depending on the context, but taking solution minus believed solution and seeing if it is or is not zero after checking the work yourself may be acceptable). However, if you are writing a paper then you could run your calculations past at least one collaborator, advisor, or peer if not a computer solver.

A: Write down what you're thinking. Don't be afraid to write down intermediate steps, even if they may seem obvious. The smaller the steps you take, the easier it is to verify that they are correct.
Try to do as little as possible in your head. You can get distracted and you can forget things. Paper does not get distracted and also does not forget things. Your mind can play tricks on you and you may remember things incorrectly. But, if you write it down, you don't need to remember it, and therefore, you can't remember it incorrectly.
In short, take small steps and write all of them down.
