How does one improve their exam performance and avoid "stupid/careless" mistakes?

Question

I have this problem in most Mathematics courses (in fact the only course I haven't had this problem in is Linear Algebra for some reason) but I study really hard, practice, and make sure to understand every concept deeply instead of just memorizing ways to solve certain types of problems but when doing an exam (note that I usually find the exam easy and familiar as it would be similar to exercises done in class and in the book) my performance is very bad usually, I do stupid/careless of mistakes due to answering with what's on top of my head instead of thinking well, I don't pay attention to certain things, and I have trouble proof-reading my exam well, I think that happens mostly due to my anxiety as I solve most things correctly during practice and they would be similar level and style, any advice on how to improve this and translate my actual knowledge into grades?

Answer 1

How does one improve their exam performance?

For improving exam performance I suggest you do practice problems but timing it. Timing is very important as it practices under time pressure which will help with performance gradually over time. Also do a lot of practices.

Avoid “stupid/careless” mistakes

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Answer 2

Making computational errors is inevitable, both within and without the context of an exam.

The simplest way to address these errors is to check your work: after solving a problem, we go back and review the solution. Therefore, one should take into consideration the additional time needed to perform this check during a timed examination. When I say "check," I don't just mean re-reading your work. I mean using specific strategies to ensure the result you obtained is correct, such as

1. substitution of the solution into the question to see if it is consistent
2. checking boundary cases
3. see if the result is the correct order of magnitude
4. try to use a different approach to the solution to see if you get the same result
5. go through your reasoning backwards
6. eliminate extraneous solutions

This is by no means an exhaustive list, and not all are applicable to every problem. The point is, part of doing mathematics is not just about having the knowledge to answer the question; it is also about having the ability to review your work critically, in such a way that you can confirm the logical steps you took are correct.

In an exam setting, my preference would be to do this checking after most of the questions are answered. If there are many remaining problems, checking may be of limited use. Moreover, I am more likely to discover an error if I let some time pass between the initial write-up and the check.

Outside of what one can do during the exam, I would say the most effective approach to avoiding errors is through practice and experience. Solving many problems, then going through and reviewing where you tend to make mistakes, is important in developing mathematical and computational proficiency. This also involves fostering good habits with respect to problem solving. Don't take shortcuts. Write down every step. Do not make an assertion or claim unless you are absolutely sure it is correct and you can justify it. All too frequently, I see students write things that are completely wrong, because they skipped several steps and were too lazy to write things out. Finally, do not blindly rely on formulas. Many mistakes are the result of improper application of formulas, identities, or theorems due to over-reliance on memorized information rather than logical deduction.

The above is just a few suggestions from a much broader picture of how to approach problem solving effectively.