My teaching experiences mainly fall into one of two categories.
In one, after I explain a concept, the student grasps it beautifully, almost immediately coming to an understanding. It's as if I only needed to provide the words that linked some unknown ideas together. This is very common with intuitive concepts, like how gravity is a force that pulls all objects down toward the ground.
In the other category, the student fails to understand the concept even after multiple different explanations. I try to adjust my wording, or use pictures, or apply the concept to real-life examples, but the student remains unable to apply this knowledge in more complicated situations. They may get a surface-level understanding, enough to follow and memorize a set of steps, but they struggle with extending this knowledge and also forget it easily.
Rarely does it feel like a student is in the middle, where I just need to hit on the correct wording or delivery format to break through to them. It may seem that way, though. The student might respond much better to a particular method because it is familiar to them, or because it is so simplified that they can mimic it immediately. But they still cannot apply it to more complicated situations and soon forget.
Why does this happen? I think the main difference comes from the preparation level of the students.
In the first, the student already has all the necessary components to understand the new concept. Simply linking the components in an explanation is thus enough for them to learn the new concept: in some cases, they can even build those connections themselves with a bit of prompting. However, in the second, there is something missing in the student's understanding of the components. Even though they are linked together by the explanation, they don't have the familiarity and intuition to fully grasp the new concept. It's like trying to build on top of an unstable foundation.
The remedy for the second problem, in my opinion, is simply to do a lot of practice. Familiarity comes from experience, and repetition is (obivously) the fastest way to get more experience. Even if the undertanding is shaky at first, just having the components memorized can build intuition over time. Once the fundamentals are solid enough, one day it'll finally "click" and all the explanations will suddenly make sense.
In retrospect, this division makes sense. If someone is struggling with algebraic manipulation, then it doesn't matter how you try to teach them the method of completing the square. They won't be able to see how one expression is equal to the next, or why the expressions should be manipulated that way. But once someone is comfortable with that, the method of completing the square is an extremely easy explanation and the student can replicate it in a variety of situations, with some prodding.
So sometimes, I think I should step back from trying to help someone and focus on directing them to improving their basics first. Although it may be a failure on my part to convey the information in a digestible way, I think there are cases where it is out of my ability to help at their current level. This goes for self-learning too: whenever I've noticed that something seems to never stick in my mind, I've started exploring the necessary components and practicing them first. I've found this to be much more effective than trying to find yet another resource that explains it a different way.
I have a lot of thoughts related to this and public education, but that'll be in another post.