I’m hoping to change this, not for all high school students, but for those I meet. I’m majoring in pure mathematics with a teaching option right now. My current plan is to teach high school algebra and calculus, as well as physics.

The other comments are right about terminology. I have no idea where the word coefficient comes from, yet I use them every day and understand them pretty well. At the moment, I’m studying ring theory, ideals, quotient rings, and ring homomorphisms and isomorphisms. Trying to figure out where these abstract terms come from may be interesting but it doesn’t help me in my proofs at all.

If you want to understand anything in math, the best approach is to read the definition, read some basic theorems, look at a few examples, and then start playing around with these things to get a feel for them. Try to prove some of the basic properties of the object you’re working with in terms of the axioms. Keep going, proving larger and more complicated theorems, and try some exercises that apply this knowledge to solve counting problems or measurement problems. If you’re learning about isomorphisms then look at some examples and try to understand how classes of object relate to one another, for example by having the same structure in their operations.

If you’re not able to understand 3blue1brown’s videos then you might benefit from using Khan academy to build up your understanding of the fundamentals. It has material stretching from grade 1 to college. Solving problems feels like playing a game, so it can be fun to sit there for hours just doing the problem sets.

Math is a deep, deep subject but it doesn’t require memorization. It just requires lots of effort and struggle in order to develop mastery.