[toogan]

The key ingredient in a formal course is the formal examination. Be it graded exercises or a written or oral exam. It forces studying at a level where you can reproduce and explain the key concepts and results and apply them to something new.

In theory one can do that also with self-study. Most people don't, they just watch some youtube video or read a Wikipedia page, and then they think they have understood something. But the deeper understanding that comes from applying this new knowledge is missing. That step is forced when you take a university course that has some form of examination in-built. Doing it yourself is possible but non-obvious, hard, perhaps even unpleasant, and it's rare people do it. Some do though, and their understanding isn't inferior to a university graduate.

[panda-giddiness]

Just to add some more motivation: in a typical physics undergraduate curriculum, you will spend roughly as much time doing homework as attending lectures. If you skip the exercises, you are quite literally skipping half of the education.

[FredF---]

It's very easy to let your brain fool yourself in understanding maths - after staring at a statement for a while, you start to think you understand it, chances are you don't.

[wolfi1]

it took me 15 minutes once to understand one equation. It was the derivation of the Taylor series for the tangent function. It was a very old textbook for calculus (from the beginning of the 20th century) and it applied techniques which are not teached nowadays, so it is possible to understand mathematical expressions after staring at it ;)

[ezconnect]

If there's a method to feel the pressure of classroom test or the requirement of finishing a project would really help home DIY learning.

[uoaei]

Unfortunately it's really hard to put the same kind of accountability on yourself compared to that of a professor gatekeeping an accredited degree. Personally, arbitrary goals don't really motivate me as the acknowledgment of its arbitrariness overrides any pursuit of reward.

[MikePlacid]

It’s entirely possible that creating a brain capable of controlling itself is more costly (in an evolutionary sense, measured by the number of generations needed to achieve this goal) than equipping a brain with the ability to check itself by communicating with others.

Nevertheless, some brains lack even that ability, gravitating instead toward echo chambers where everyone shares the same views, so no mutual checks are possible.