This reminds me of a discussion in the most recent episode of Django Chat with Aymeric Augustin on the difference between tutorials and reference documentation.
How to Understand Things
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Re: How to Understand Things
#22> But it’s not just energy. You have to be able to motivate yourself to spend large quantities of energy on a problem, which means on some level that not understanding something — or having a bug in your thinking — bothers you a lot. You have the drive, the will to know.
This resonates with me. Someone once asked me how I decide when I'm finished with a particular thing I'm working on. The answer is as simple as "when I stop thinking about it". When it stops bubbling up in my thoughts. Until then, I'll keep returning, and I'll keep chipping away.
Re: How to Understand Things
#23As an engineering manager, I love having a mix of people who just always need to go deep on whatever they’re working on and others who are just obsessed with shipping and getting stuff out. Particularly great when you pair them off and they push/pull each other a bit.
Totally agree with this. I used to be solely in the latter camp and my teammates pulled me in the deep direction.
Re: How to Understand Things
#24How would you create a training program to teach all of lessons?
I thought of creating a workshop at my uni, titled "how to ask stupid questions?" Essentially, do group activities where someone presents on some topic, and the goal of the audience is to ask genuine "stupid questions" - questions about the fundamentals, which most people are embarrassed to ask, but which play a big part in understanding.
Re: How to Understand Things
#25Jesus christ that background is distracting.
Re: How to Understand Things
#26> This quality of “not stopping at an unsatisfactory answer” deserves some examination. > This requires a lot of intrinsic motivation, because it’s so hard; so most people simply don’t do it. It also requires self-confidence, persuasiveness, and social power. Without these traits, your attempts to really understand something will be dismissed as "overthinking things" or "trying to understand the universe". Those arou…
It’s worth mentioning that this desire to think through things deeply often runs contrary to a bias for action, and it’s very difficult to measure progress in thought. There is a long phase of thinking very hard, and a short phase where things suddenly become clear. So, thinking through and understanding things deeply is often discouraged as a consequence.
Re: How to Understand Things
#27Jesus christ that background is distracting.
Wasn't sure whether your comment was relevant, then I remembered I closed the article a third of the way through because of a deep annoyance with the background.
I'm all for individual expression, but here I think it subtracts from the reading experience.
Re: How to Understand Things
#28Great post. Personal anecdote: I don't think I really understood anything in school, but I was decent at going through the motions of carrying out certain methods and recalling certain facts when I needed to. I went on to study Maths at university, and for most of my first year, I had the same surface level "methods + facts" knowledge that got me through school. After some studying, I could recite definitions and the…
OP here -- that is a fantastic anecdote! Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)
As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example.
As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might even have remembered some verbatim proofs of some of the equivalences.
However, I developed absolutely no intuition about what a determinant is. I had book knowledge, but no insight. It was a long time ago now, but I’m fairly sure that when I graduated I still did not truly understand even this very basic (by undergraduate standards) subject. I think it was probably a few years later, when I came across some of the same theory but in a much more practical context at work, that most of the connections in that equivalence theorem first “clicked”.
Meanwhile, here is what a gifted presenter with the right illustrations can do in about ten minutes:
https://www.youtube.com/watch?v=Ip3X9LOh2dk
The 2,000 or so substantially identical comments below that video are very telling.
Given the understanding you’d get with that quality of presentation, the equivalences I mentioned above would have been obvious and constructing the proofs from first principles would have been straightforward.
Re: How to Understand Things
#29Earlier quoted context omitted.
OP here -- that is a fantastic anecdote! Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)
As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example. As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might ev…
Awesome example, thank you!
Re: How to Understand Things
#30Not stopping after the first right answer is one metacognitive strategy[1] among many. Metacognition is an area of active research.
I first heard of metacognition as a distinct discipline in connection with the treatment of insecure attachment.[2]
Metarationality[3] is an aspect or extension of metacognition.
[1] https://helpfulprofessor.com/metacognitive-strategies/
[2] https://www.psychologytoday.com/us/blog/the-resilient-brain/...