Learning (autodidacting)
Recognise dependencies. Make things easier and more palatable. Give time for foundational level to solidify in the unconscious mind, before moving to advanced level.
Context:
Thought 1:
Look for a gentler learning curve.
A difficult, complex skill can often be decomposed into easy, simple sub-skills. Or can be learnt via starting with a simpler version.
Read Easier Textbooks Instead of Struggling Valiantly
TL;DR even though slogging through tough textbooks makes you feel sophisticated and smart, don’t.
Imagine I’m learning how to program, but I’ve never used a computer before. Learning to program while learning to operate a computer, will probably take longer than learning to operate a computer and then learning to program. Learning time is superadditive in terms of your ignorance / the dependencies you’re missing.
But it’s worse than that. Imagine I’m learning quantum mechanics, but I don’t know any linear algebra either. I’m now trying to do three things:
Learn linear algebra,
Learn the formal postulates of quantum mechanics, and
Tie all of this into the real world.
… …
Crucially, via this method, I’m only confused about one thing at a time. Build models in the right order!
Related: « Learning: “good stuck” vs “bad stuck” »
https://jacob-tan-en.medium.com/learning-good-stuck-vs-bad-stuck-8957773e03ef
Thought 2:
Mathematical concepts build on top of sub-concepts. Sub-concepts take time to solidify. Rushing on to the next concept before the relevant sub-concepts have solidified, is counter-productive.
A better alternative is to study multiple topics (that are not directly related to each other) in parallel, instead of doing it in series.
I like the multi-layer cake analogy. E.g. Calculus 2 depends on Calculus 1, and Statistics 2 depends on Statistics 1. After reading Calculus 1, the concepts take time to solidify (bake) in the unconscious mind (oven). Rushing to Calculus 2 immediately after, would be like taking the cake out of the oven and putting on the next layer, before the base layer has properly solidified. Instead, one should read Statistics 1 next …
Read Several Textbooks Concurrently
TL;DR study several topics at once so that your brain has time to cement the concepts you’re learning, before the text builds on those concepts further.
AllAmericanBreakfast’s recent post is great, so I’ll refer you to that to make this point:
Wait, what? You want me to make life easier on myself by, instead of studying calculus…studying calculus, linear algebra, and statistics all at once?
~ AllAmericanBreakfast, The Multi-Tower Study Strategy
The basic idea is that your brain needs time to really cement a new idea in, and so you should study several topics at once in dependency-heavy areas like mathematics. This advice matches up with both my recent personal experience and with advice I received early on from Qiaochu Yuan, but which I had unfortunately ignored.
Extra reading:
