In mathematics students have to grapple with problems to understand it. It’s only when they go through the effort of investigating a problem on a deep level that they understand it.
Solving for x doesn’t mean much when all the steps are written down in advance. Just copying answers is useless. But mentally grappling with the problem—that’s where the gold lies.
Really grappling with a concept might mean that you don’t get it at first. It might mean that you completely misunderstand it initially. It might even mean that it takes you days to catch a glimpse of what it means on a practical level. But asking questions, pondering, contending with confusion, and testing procedures—that deepens understanding.
To understand things well is the best way. We have to leave the monkey-see-monkey-do approach behind. But students like to be taught, meaning they like to sit and listen while you explain something. When you encourage them to think, they resist. But not letting them think, cripples them in the long run.
Letting students grapple with ideas and problems is the way to mastery. Ironically, this doesn’t require a lot of effort from a teacher. Students can teach themselves. This does, however,require an iron will from the students to not get distracted. It requires deep focus. Not understanding something can be supremely uncomfortable. The trick is to … well … suffer through the discomfort.
This is good news for those that can’t afford expensive degrees. Just train focus. Read a lot. Re-read until you understand every nuance. Don’t just give up after the first try. Try ten times if you must. Ask questions. Attempt sample problems. Grapple with the material. Don’t be afraid.