My trick is to find the paper you want to read, but immediately skip to the references; recurse until you get to a paper that you more or less understand.
It’s a bit time consuming but it makes paper reading a lot more fun.
“My biggest mistake when starting my doctoral research was taking a top-down approach. I focused my efforts on a handful of research papers on the frontier of my chosen field, even writing code to solve problems in these papers from day one. However, I soon realized I lacked many foundational prerequisites, making the first year exceptionally tough. What I should have done was spend 3-6 months dissecting the hell out of all the key research papers and books written on the subject, starting from the very basics (from my knowledge frontier) and working my way up (the bottom-up approach).”
I would legit PAY for an app that managed dependencies for understanding a paper so that I could see the path I need to take to understand what I'm reading. Could apply to books too.
perhaps, with the advent of AI, one will be able to convert a current book into a more detailed book, and also a current book into a smaller book, so maybe this idea is even easier to implement than 4 years or so ago, before chatgpt (but after summarizers, which prompted the idea in my head).
I would humbly appreciate any feedback on the concept of parametric books, for now, it's just an idea, but it's a free one, anyone is free to implement it.
This is one reason why it's helpful to learn bottom-up when possible as opposed to diving straight into the deep end and trying to fill in missing knowledge as you go.
Hm, I've always felt bottom up more difficult to learn. I always found it helpful to first have an overview, a mental map of sorts, of the high-level details, so that when I looked at the details later I could make connections and know where to "put" this knowledge relative to other things.
With bottom up I always feel lost because I don't know what it's useful for, the relationships to other pieces of knowledge, etc.
It can definitely be helpful to take a top-down approach in planning out your overarching learning goals.
However, the learning itself has to occur bottom-up. Especially in math. Math is a skill hierarchy, and if you cannot execute a lower-level skill consistently and accurately, you will not be able to build more advanced skills on top of it.
It's good to have a high level view of what your ultimate goals are, but if you are lacking too much foundational knowledge you can't even conceive of it. Especially in a subject like math, everything builds from the bottom up.
We don't give first graders an overview of differential equations and their applications when we start teaching them addition and subtraction.
I think there are certain circumstances where getting in over your head and digging your way out is a better approach -- but I don't know how to distinguish those cases from the rest.
I don't think there's anything wrong with trying to jump headfirst into things that interest you. I would just recommend that you need to be honest with yourself about whether you're making progress -- and if you're starting to flail (or, more subtly, doubt yourself and lose interest), then it's an indication you need to re-allocate your time into shoring up your foundations.
Papers are a terrible way to learn unless you are already an expert in the field, because the prerequisites tend to be "the entire rest of the field". It's a rare paper that actually assumes you might not know everything the authors knew.
