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How to Understand Things

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Re: How to Understand Things

#41

Post by knzhou »

alicemaz wrote: post #13
refrigerator wrote: post #6

Great 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…

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil

What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confused word salad. People waste years of their lives this way, going in circles without ever carefully building their understanding from the ground up. The order in school was chosen for a reason.

Re: How to Understand Things

#42

Post by refrigerator »

Chris_Newton wrote: post #28
nqureshi wrote: post #8

Earlier 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…

This is awesome, determinants were one of the things that I never really understood during my degree

Re: How to Understand Things

#43

Post by cmehdy »

inetsee wrote: post #17

One of the things I remember quite vividly from my Psychology classes in college was the idea of a "satisficing" problem solver. Given a (reasonably solvable) problem to solve most people can come up with a satisfactory solution. The difficulty comes when you ask them to come up with a new solution. Many people struggle because their brains say "I already came up with a solution, and it was a pretty darn good solutio…

To me this seemed to get easier as I encountered more "languages of the mind", i.e. more ways to think about things. This, is very much tied to the "nurture" part of our lives, as foundational experiences are more by the very definition of "experience" subjective and unique.

Solving a problem within mathematics in a new way can be made much easier if you have grasped multiple fields (for example, algebra vs. geometry). I've seen people understanding chemistry well because they enjoyed cooking, and could sort of use either to get to a given explanation to solve a problem. I've definitely started to grasp chemistry only when I reached a decent level in theoretical physics.

Here's my assumption: everything has a likelihood of depending to some degree on other things (examples: can you do mathematics without a language or writing? can you do physics without mathematics?). Therefore, "thinking laterally" could very well be thought of more as "thinking with an interesting combination of previous vectors of thought". Perhaps the "genius" is to create nonlinear combinations of previous vectors.

So in short, this ability to come up with another solution, and another, and another.. I wonder how much it is tied to the richness of experiences you've had since your birth, and particularly the foundational ones (at the very least I would assume to be more testable than later on, if my hypothesis about dependency of "thought vectors" is true).

Re: How to Understand Things

#44

Post by pc86 »

knzhou wrote: post #41
alicemaz wrote: post #13

Earlier quoted context omitted.

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confu…

It didn't work for the GP. That makes it poisonous and evil.

I've seen this sentiment way too much on HN. X didn't work for me, therefore X is a scam, its perpetrators are evil sociopaths, and if it worked for you you're a cog in the machine, man.

Re: How to Understand Things

#45

Post by cloudier »

pstuart wrote: post #11

Interesting. tl;dr -- we learn by experiencing things, not by being told things.

I think this is the transmissionism versus constructivist view on teaching.[0] This is something that is well-known in education and I wish more people knew about it! Lay people commonly think of education as transmission of ideas from teachers to learners, but educators believe that learners construct their own understanding of ideas. So these educators try to create situations where the learners can do that construction.

[0]: http://nas-sites.org/responsiblescience/files/2016/05/Dirks-...

Re: How to Understand Things

#46

Post by philjr »

ssivark wrote: post #26
afarrell wrote: post #20

> 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.

The key is to decouple the two.

When you are required to act, act and act decisively. If you are clear that the understanding could be deeper (and it usually can), you trigger a work effort to understand more. So the next time you need to make a decision you’re more informed.

Re: How to Understand Things

#47

Post by djohnston »

Excellent read. So much of this resonates deeply with me based on uni experiences. I studied math and was guilty on more than one occasion of basically memorizing some theorems and methods of manipulation, missing the forest for the trees so to speak.

Re: How to Understand Things

#49

Post by BeetleB »

knzhou wrote: post #41
alicemaz wrote: post #13

Earlier quoted context omitted.

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confu…

I suspect you and GP are talking about slightly different things. GP is probably more opposed to artificial compartmentalizing of things. As an example:

> You do need to understand calculus before classical mechanics

Yes, but how much calculus? Do you need all of Calc I, II and III before even attempting classical mechanics? And should calculus even be treated independently of classical mechanics?

There are various traditions when it comes to teaching these subjects, and the tradition in the US involves keeping a strict distinction between these things, in addition to a "theory first" approach. Other people have studied things in a different manner. Some of my physics professors from the UK had studied most of the math they knew only as needed when they would get to relevant topics in physics - including differential equations, all of analysis (complex or real), some of Calc III, etc.

Even amongst mathematicians, it was common in parts of Eastern Europe to focus on a problem, and learn whatever theory is needed to solve that problem. They didn't learn theory and apply to problems - they took a problem, and learned whatever theory is needed to solve them. I recall picking up a Kolmogorov textbook on analysis and being surprised by seeing this approach, along with the informality with which everything is discussed.

And just a minor quibble:

> classical mechanics before quantum mechanics,

You don't really need to know much except the basics. I think the classical mechanics we covered in our typical "Engineering Physics" courses was sufficient to dive into proper quantum mechanics. It's nice to have been exposed to Hamiltonians in classical physics prior to taking QM, but really not needed. There's a reason neither schools I attended made the classical mechanics courses as prereqs to QM. In fact, I would argue we should split things up a bit: Have a course to teach the very basics of energy, momentum, etc. Then make it a prerequisite to both classical mechanics and quantum mechanics.

Re: How to Understand Things

#50

Post by drorco »

knzhou wrote: post #41
alicemaz wrote: post #13

Earlier quoted context omitted.

>I suspect lots of people go through the entire education system like this. +1. it took me a couple years after getting kicked out of college to get my head sorted out to the point where I felt like I could "think" again I think one of the most harmful things about schooling is the way it imposes a tracked structure on learning. it demarcates knowledge into discrete subjects and sets up a linear progression through t…

> says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil What’s wrong with it? You do need to understand calculus before classical mechanics, classical mechanics before quantum mechanics, quantum mechanics before quantum field theory, and quantum field theory before the Standard Model. I’ve seen tons of people disregard this and the result is always confu…

Some people like to work their way in reverse. I'd often pick a really complicated subject I'm after like "stellar fusion" and then work my way downwards and learn whatever I need to learn in order to understand it. If I had to start from differential mathematics, without knowing why I need it, I'd probably give up.
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