Because if it feels difficult, it’s likely there is something missing. Either some connections between different parts (so instead of a connected network, it feels like a list of isolated facts), or you memorized some rules but you don’t actually know why it works that way (you wouldn’t be able to re-derive those rules)
I agree that this is true but I’d like to ammend “Learn until you actually get it (or don’t learn at all)” to “Decide whether your goal is to completely understand the material or use it instrumentally to achieve something else. Then choose to learn it completely or actively optimize for understanding it instrumentally.”
In the context of mathematics, the line between “deep concept I’m missing to understand this” and “one-off clever but counterintuitive step that turns out to work, possibly for a deep reason outside the field I’m working in” is pretty blurry. As a result, when you learn mathematics, you tend to have a set of knowledge composed of Deep Things You Truly Understand and a set of memorized things composed of One-Off Clever Hacks That Don’t Give Me Much Knew Insight Into The Things I’ve Learned. I’ve found that I do best when, in the process of learning the material, I try to systematically categorize the information I learn into those two categories.
An Example:
I know a lot of things (especially things related to my applied mathematics major) that I’m extremely confidant that I don’t see as isolated facts (ie, the idea of orthogonal functions, the idea of differential equations, the idea of sines and cosines, the idea of Fourier series) and that I’m extremely confident make up the set of things I need to know to understand complicated facts (ie, solving partial differential equations with Fourier expansions). I’m even relatively confident that I could re-derive how to solve partial differential equations with Fourier expansions eventually in certain cases.
However, the number of intermediary steps I would need to solve these partial differential equations is still pretty large (ie, identifying the specific boundary conditions, deciding which expansions to try with which boundary conditions, figuring out which simplifications help things cancel out, etc.) and the way I would use my knowledge to carry out these intermediate steps is still pretty challenging and non-trivial. Re-deriving them is probably do-able but it’s not something I want to be bothered with. Learning through repitition would do the trick to, but if I abandoned the repetition for a year, I’d lose the benefit.
Turns out the best solution I’ve found is seeing a math problem, thinking “hm this reminds me vaguely of something I did two years ago”, and then searching Paul’s Online Math Notes to see if they solved a problem similar enough that I can steal from them.
I agree that this is true but I’d like to ammend “Learn until you actually get it (or don’t learn at all)” to “Decide whether your goal is to completely understand the material or use it instrumentally to achieve something else. Then choose to learn it completely or actively optimize for understanding it instrumentally.”
In the context of mathematics, the line between “deep concept I’m missing to understand this” and “one-off clever but counterintuitive step that turns out to work, possibly for a deep reason outside the field I’m working in” is pretty blurry. As a result, when you learn mathematics, you tend to have a set of knowledge composed of Deep Things You Truly Understand and a set of memorized things composed of One-Off Clever Hacks That Don’t Give Me Much Knew Insight Into The Things I’ve Learned. I’ve found that I do best when, in the process of learning the material, I try to systematically categorize the information I learn into those two categories.
An Example:
I know a lot of things (especially things related to my applied mathematics major) that I’m extremely confidant that I don’t see as isolated facts (ie, the idea of orthogonal functions, the idea of differential equations, the idea of sines and cosines, the idea of Fourier series) and that I’m extremely confident make up the set of things I need to know to understand complicated facts (ie, solving partial differential equations with Fourier expansions). I’m even relatively confident that I could re-derive how to solve partial differential equations with Fourier expansions eventually in certain cases.
However, the number of intermediary steps I would need to solve these partial differential equations is still pretty large (ie, identifying the specific boundary conditions, deciding which expansions to try with which boundary conditions, figuring out which simplifications help things cancel out, etc.) and the way I would use my knowledge to carry out these intermediate steps is still pretty challenging and non-trivial. Re-deriving them is probably do-able but it’s not something I want to be bothered with. Learning through repitition would do the trick to, but if I abandoned the repetition for a year, I’d lose the benefit.
Turns out the best solution I’ve found is seeing a math problem, thinking “hm this reminds me vaguely of something I did two years ago”, and then searching Paul’s Online Math Notes to see if they solved a problem similar enough that I can steal from them.
Yes, great correction. I’ve modified the post to state that it only applies for Deep Things You Truly Want To Understand.
I’m often confronted with the difference you’re describing but haven’t ever articulated it as you just have.