I believe this is a huge shortcoming of how math is taught. You can bet your last dollar that the teacher doesn't think it's about tricks. But the students are convinced that it is. Students and teachers are both exposed to the exact same material but end up with diametrically opposing conclusions.
Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years.
The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades.
The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works.
Disclosure: I taught college freshman math for one semester.
My high school calculus teacher was really great at this. He didn't just teach a bunch of transformations to memorize. That was part of it, naturally, as you aren't going to use the definition of limits and the FToC in all your problem solutions. He actually made us construct volumes from pieces of poster board, measure the segments and calculate the Riemann sum. When we did function analysis, he didn't allow us to use the Cartesian plane at first. We had to show visually how a function deformed the one-dimensional real line. How x^2 squished values between -1 and 1 toward 0 and stretched the other values toward +infinity.
It gave me a good "visual" grasp of the concepts and made most of my higher math classes much easier.
I do agree diff eq instruction sucks. I got an A in that class and didn't understand a thing. "This equation has this form; this is the canned solution."
Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years.
The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades.
The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works.
Disclosure: I taught college freshman math for one semester.