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The theory makes you understend why a method works for a certain problem. A lot of exams try to trick the taker by giving problems that are almost solvable with just the toolbox but need a bit extra trick to solve which theory can help. But there again i find that simply knowing the specific trick is enough to do well.
But personally believing that is in any way important to succeed in exams has lead me to waste too much time. If you find that you have prepared well enough to solve any problems across math for an exam, it would then be ok to then cover the theory.
So in essence, just keep doing what you're doing.
It’s reassuring to hear that focusing on problem-solving isn’t necessarily a drawback, as long as I’m prepared for a wide variety of questions. I think I’ll stick with my method for now and revisit theory selectively when I feel gaps or curiosity arise.