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Beyond notation and ordering, I have had the best results reading and comprehending complex concepts, including mathematics, by taking the following suggestion
by johnbender 9y ago
Beyond notation and ordering, I have had the best results reading and comprehending complex concepts, including mathematics, by taking the following suggestion to the extreme:
> Read with pencil and paper in hand, making up little examples for yourself as you go on.
I like to find a difficult question that I can answer with an understanding of the material. This acts as a litmus test of my understanding and a forcing function.
The question can be almost anything, but a general approach I use is to write a "compiler" that maps some concept from the material to a concept I already understand (this normally takes the form of a denotational semantics). Then the question would be, "How can I interpret X as Y?" This technique has its limits since the material can't be too far afield from something I already know and the idea isn't novel but it has been effective for me. The critical bit is forcing myself to write down a fairly comprehensive mapping function. This gets me into the dark corners of my understanding very quickly and adds new questions to answer.
- aisofteng 9y agoI do the exact opposite - I don't write anything down and try to visualize what I'm reading entirely. I don't start writing anything down when doing exercises until I have the entire solution in mind (unless it's something computational). It's hard to get used to doing at first, but, after a couple years, my abstract reasoning skills developed significantly past most or all of my peers.
- logicchains 9y agoI've noticed something similar; it's harder at first, but once I learn to think about something visually/intuitively, I can understand/use/reason with it much better than if I'd started with pen and paper. It makes sense that this approach is more effective because if the brain has internalized a concept than it can reason with it subconsciously, but if external paper is required for reasoning about it then such subconscious reasoning is not possible.
- compactness 9y agoAt some point math becomes more about ideas and less about lower level details. Like you I don't write down my reasoning either and try to do exercises in my head or verbally. So long as I get the idea behind the proof correctly, I am not worried about the lower level implementations.
- yakitori 9y agoThe best advice when learning math or anything is to "solve the smaller problem". Try to break the problem down into smaller pieces and work from there. Trying to understand something complex as a whole can be daunting and intimidate people from even trying to understand it. But if break it apart into smaller parts and solve/understand those, many times, you find that it wasn't that complex in the first place.