Math as a field of study might be hard but in my experience math pre grad-school is just memorization as much as any other subject. I studied for my exams by simply doing countless exercises and memorizing every problem pattern to the point where I'm just plugging numbers into a series of formulas, even if I don't fully know why I'm doing so.
This is my experience as well. And unfortunately, many of my teachers feed into this. "Ok kids, today we're going to learn this new formula. You put a number in, do some arithmetic, and you get an answer out. I am not going to bother explaining the importance of this formula, how it came into existence, nor its practical application — likely because even I don't know. But what I do know is it's important because everyone says it is and it will be on the exam. Now get cracking." A parody of course, but I say captures the sentiment which completely drained my desire to learn math. I got good grades, but only after I learned to accept that it is futile to learn the importance of what I am learning and instead simply focus on rote memorization of solving the problem, even if I don't know why it works.
> Ok kids, today we're going to learn this new formula. You put a number in, do some arithmetic, and you get an answer out. I am not going to bother explaining the importance of this formula, how it came into existence
Short story time: I never could memorize various geometry equations, like surface area or volume of basic shapes. Then one day while bored at my part time job, I found a paper and pencil and decided to use what I'd just learned in calculus to see if I could derive one of those equations, by leaving variables in instead of using concrete numbers.
It totally worked, I reinvented the volume of a sphere equation and ever since that day I've never forgotten it because now every part of the equation has meaning. I know why it is the way it is, and it makes sense.
This gets to the heart of what math class needs. Which is a process of learning how to re-invent formulas, constants, and other concepts. Through this, students will come to appreciate why such things exist. And they will appreciate it further if students are first challenged to solve such problems in the absence of it.
There is truth in this, but even then, to apply the formulas to many problems, you need to have seriously strong pattern recognition abilities, and the capacity and flexibility to adapt what you know to the given situation. Then piping the right thing into the other right things requires a deeper understanding, not always of what the thing very nature, but at least what it does, and what's it compatible with.
This means loading the entire problem space in your head.
