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 theorems, I'd memorised some proofs, and I could occasionally manipulate a problem to get an answer. I think about half of the cohort was in a similar position. But it was clear that there were others in a completely different league.
When we were studying for our first year exams, I was struggling to remember the proof of a specific theorem (it felt quite long). A friend was trying to help me learn it, and he asked me what "picture" I had in my head for the theorem. I didn't have any pictures in my head for anything.
It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive. It was long-ish to write out in words, sure, but the underlying concept was really simple. This blew my mind — I realised I didn't have a clue what we'd been studying the whole year.
The worrying thing is that I actually thought I understood that stuff. Before that incident, I didn't know what it feels like to actually understand something, and I didn't have an appreciation for the layers of depth even within that. I suspect lots of people go through the entire education system like this.
> It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive.
Excellent point.
1. A similar example: Feynman diagrams.
2. Another: Venn diagrams.
3. Longer example: On Navy nuclear-powered aircraft carriers, the officer of the deck underway (OOD) must have at least a basic understanding of how the engineering plant works. It's second nature for nuclear-trained OODs, of course, but non-nukes could sometimes have trouble. Back in the day, it turned out that an effective way to help non-nukes learn what they needed to know was to have them: (A) memorize a really-simple block diagram of the reactor and steam system, and also (B) memorize a chant, of modest length, that summarized how things worked. During slow periods while standing OOD watch, I'd make a non-nuke OOD trainee draw the diagram from memory; then I'd quiz him with "what if ..." questions (back then it was always "him"). If he got hung up on a question, I'd tell him, "chant the chant." That usually helped him figure out the answer in short order.
(U.S. submarines don't have that problem, AFAIK, because pretty much every officer who will stand OOD watches is nuclear-trained.)
>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 them and says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil, and I've encountered many people who are crippled for life by it. a lot of people never pursue things they're really interested in and could become extremely passionate about because school has convinced them they need to stack up prerequisite knowledge before they're even allowed to touch it
Our school is poisonous (can tell for France), if not evil. It become crystal clear after reading Celine Alvarez. Not sure if she got translated yet. In english, but older you also have Alfie Kohn, but I haven't read him.
When reading Celine, one understand that children are natural born learner, and there is no effort needed to make them learn stuff. Our school model is industrial production of objects. Thinking human machines. We are way more than that. Sadly Pink Floyd description of the school still echo to our modern school. Some peoples don't feel that way about school. I don't really know why. Maybe they never imagined how better it could have been, so they found it great.
> 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.
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:
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.
Ahh I did a bit of googling but couldn't find anything nice — sorry! Most of the time, the complex stuff is broken down into smaller "lemmas" with their own manageable proofs, and then the proof of the whole theorem will be something like "Follows from Lemma 2.1, Lemma 2.2, and a basic application of Theorem 1.4"
Yes this.
In grade school I had trouble memorizing the formula for the area of a right angled triangle.
But one day I bisected a rectangle diagonally and was blown away how it all of a sudden made sense:
The same thing happened with circles and even calculus stuff.
Putting it into diagrams suddenly made it make sense.
So now I try to “do” instead of memorize and it makes all the difference in the world.
Even I can’t learn new programming languages unless I make a real project with them. That’s why I don’t believe in things like code katas or reading programming books cover to cover. It’s a bad representative for using a tool in real life to learn its principles, techniques and methods.
And if you got good grades, then the system wasn't testing whether you understood it - Bad news.
But it's good news that you were able to understand once he drew the picture for you. So there is an effective way to teach that theorem - If only the professor knew it.
I felt the same exact way trying to memorize molecular formulas in organic chemistry, being able to visualize the molecules and some outlandish pneumonics really allowed me to force my brain to make memories that I forgot over time however I gained the ability to "see" the molecules in my mind and deriving the formula became trivial.
Creating really unique pneumonics works similar to how you can recall when something happened out of the ordinary or have a "sixth" sense where you subconsciously can sense something is off in an otherwise ordinary situation or encounter. If I had to guess, this might be an evolved human survival mechanism that helped our ancestors detect and avoid dangerous/deadly situation.
I wonder what other evolutionary mechanism we can hack to encode concepts more deeply. I know spaced repetition is the mainstream approach via flashcards/ankii type apps. I think pneumonics blended with spaced repetition is the most practical methodology. Thoughts?
And _why’s poignant guide! I’m still a rank amateur, but that little PDF helped me truly understand, not just memorize.
Humans are wired to enjoy and remember stories. So much education is done narrativelessly and as such neglects to harness one of our most potent capabilities.
What happened after this revelation? Did anything change? Did you try to intuitively understand the problems you were working on? I think this boils down to how you were taught (or self taught) to approach the subject and the set of tricks you learned along the way that became your toolset.
Did you do a lot of exercises in school? They usually help build the intuitive part, the aha moment, that comes through repetition.
Yup, definitely changed the game for me, and turned on my 'intuition' spidey sense, of whether I actually understood a concept.
We did do a lot of exercises in school, but they mostly just tested whether you can reliably apply a method that you were taught, rather than testing understanding.
Haha, yeah. Thankfully, the Maths exams were set up such that you could always get 60% (the boundary for a "2:1" grade in the UK — an acceptable score for most people) by just knowing the 'bookwork', which you could do by memorising stuff without understanding it.
The university I went to offered open book/note exams for almost all courses. It literally didn't even matter how much you memorized... open book tests didn't make anything easier. you need to understand or fail.
I'm not into showing off ranking or pedigree but I do genuinely believe that the higher the pedigree your school the more likely the exams will be harder and require total understanding and even creativity over rote memorization.
The reason is because memorization is trivial. Students able to get into any top school will likely all easily achieve full score on an ordinary tests. The professors at top schools need to make these tests brutally hard in order to produce a bell curve.
I literally had one new professor at my school actually give a mid term that was what would ordinarily be called fair in any other school or college... but the entire class ended up getting nearly full score.
He realized his mistake and the final was way, way harder.
Rote memorization tests are the easiest to make and grade, both in terms of simple instructor effort as well as in terms of fairness and lack of ambiguity.
In Asia it is common for schools to solely grade base on multiple choice exams which promote cramming and memorization. But in the US I would be shocked if a program that only operated that way was accredited. I would argue students who simply memorize things are trying to hack their way to an unearned grade. The goal of higher education is supposed to be to reach higher levels of Bloom's taxonomy, beyond the rote stuff done in primary and secondary school.
> Students able to get into any top school will likely all easily achieve full score on an ordinary tests. The professors at top schools need to make these tests brutally hard in order to produce a bell curve.
This is no longer how “high pedigree” schools function in the US. There is too much blowback from helicopter parents and the kids who haven’t ever had to deal with critical feedback before.
Lookup “grade inflation” if you want to be horrified by the quality decline of “pedigree education”.
> 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 around you will urge you to "stop thinking just do the task" or "do the obvious thing" as they lose patience with you. If you don't resist them you'll end up moving forward despite feeling confused, sometimes completely. You'll then end up pissing people off when you execute too slowly or fail (in their eyes, intentionally).
> This is a habit. It’s easy to pick up.
Not if the people around you are exhausted by you.
I want to add - habits like that are great to pick up, and fine tune, when you're in college.
More so if you're a college student that can focus 100% on your academic life/studies, unburdened by things like work.
EVEN more so if you have great mentors, professors, etc. that can guide you to the right place.
I'm not saying that one CAN NOT do the things above, if you're a busy student with work on the side, and very limited resources as far as mentors or professors go...but I do think that those lucky enough to find themselves in the right positions, are more likely to mature - and quicker.
(And it was no surprise to see that the author is a PPE grad from Oxford)
Your points about social context are great ones and 100% valid, but I want to make a case for "stop thinking just do the task":
---
Often, trying (and potentially failing) to do a task is the best way to learn about it. The key is to be very explicit about what parts of that task you actually understand and which ones you're pulling out of your ass.
This is especially true when creating software. It's super rare to have requirements that are concrete and detailed enough to form a comprehensive understanding of the final design before you start developing it. Instead there are usually parts that have clarity and others that are fuzzy. If you can enumerate those and keep them separate you can often leverage the parts you understand to make progress on those you do not. Writing placeholder/obviously-terrible code to stand in for the unknown parts just so that you can spin up a running system is a great way to do that. Along the way you'll see what patterns emerge, where you hit walls, etc, which is not always easy to imagine with raw abstract thought. And having "working" software that you can play with is a great way to find edge cases and otherwise make progress on those unknowns. Once you've gained a more complete understanding you can replace the placeholder junk with well-designed/actually-thought-out modules.
(Another obvious reason to do this is if your company will literally go out of business if you wait until you have a perfect understanding to launch a product/service, but I think most people here get that.)
I'm not sure how much this generalizes, but it also works well for me when writing. I usually start with a vague understanding of an idea I hope to communicate, then jot down disjointed sentences to capture parts of it. As I do so it gradually becomes clear how things are connected, where my reasoning is muddy, what I thought I knew but can't express so probably don't, etc, and I can use this gained understanding to iteratively rewrite and reshape my message until it becomes something coherent. Sometimes, anyway; other times I don't end up sending/publishing it at all because along the way I learned that the thing I was hoping to communicate was based on a faulty assumption or is not as straightforward as I thought it was. Which is great, because either way I've learned something.
---
I guess I'd say it differently: "keep thinking and do the task".
Agree with this. I think of education as a "series of increasingly smaller lies". If you try to explain everything right away with complete fine-grained truthfulness, you quickly get to molecular dynamics, and don't have to stop there.
A common unspoken theme in the post is, the art of learning is knowing "what" to care about and "when".
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.
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.
There are both extremes, and best not to be at either one of them.
Example: I forget the name of the principle, but in mathematics the statement "P implies Q" is considered true if P can never be true. As an example, let P be "George Washington was a woman" and Q be "Queen Elizabeth is a man". Then the statement "If GW was a woman, then QE is a man" is considered to be a true statement.
I have a friend who refuses to accept that such a statement should be considered "true". And he has put off studying real analysis until he can learn enough logic theory to convince himself on the validity of accepting such statements as true. I do not think he'll ever get to study real analysis, because he is full of "No! I need to understand this really really well before proceeding!" statements.
It's a fine approach if you have an infinite amount of time.
The other issue, as another commenter pointed out: It's very difficult to measure progress in thought. The mind is great at fooling itself, and not until you try to solve real problems (or discuss them with others) will you expose most of the gaps in your mind. The same person in the above anecdote does suffer from this. He definitely puts in effort to learn a lot (and has succeeded), but there are always more things to learn, and he moves on to the next topic before really applying what he has learned. As someone who talks to him often, it's really hard to tell if he understands. He is the classic case of "I'm sure I can solve problems when I need to, with a bit of review".
At the other extreme, of course, are people who are not really that motivated to understand. They are satisfied if they get the answer at the back of the book. You won't get far with just that.
> If you don't resist them you'll end up moving forward despite feeling confused, sometimes completely. You'll then end up pissing people off when you execute too slowly or fail (in their eyes, intentionally).
This sounds more like an issue at work, and your experience is fairly universal - most jobs I've worked at have it. In my experience, understanding things well is sadly not valued on the job. They want you to "execute", and want you to minimize the time you spend learning. And of course, they would rather hire someone else instead of ensuring your proper learning/training.
A real logician can correct me if I'm wrong, but I think those statements are considered "vacuously true" for the same reason we say the empty set is a member of every set and x^0 is 1: it makes the rules of calculation simpler if we define it that way than other ways. But you can't use these convenient definitions to learn anything new.
You can keep multiplying your equation by x^0 and union-ing your set with the empty set and or-ing your proposition with "If eggs are diamonds then fish can talk" all the livelong day, but you never gain any more information.
> the statement "P implies Q" is considered true if P can never be true. As an example, let P be "George Washington was a woman" and Q be "Queen Elizabeth is a man". Then the statement "If GW was a woman, then QE is a man" is considered to be a true statement.
> I have a friend who refuses to accept that such a statement should be considered "true".
This is a weird thing to refuse to accept, since it is an ordinary part of vernacular English, a common way of dismissing an assertion as false.
"If that guy graduated high school, I'm the King of England."
Actually there were women who dressed like men, pretended to be men -- and if it, surprisingly, turns out this was the case with GW, then that won't make QE a man.
Yeah, there's a bunch of caveats here. You also need to be reasonably quick to understand things, otherwise you'll be spending a lifetime trying to understand some pretty basic concepts and never get to anything interesting.
In general, there are a bunch of starting conditions necessary for this kind of approach to glean value.
I don’t agree because I still do this all the time. Sure some people get annoyed, assholes will try to make you feel bad for asking dumb questions, or even mock you publicly. The fact is, your understanding grows so quickly when you ask questions that this doesn’t matter.
As 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.
I have always thought that, as mentioned in the article, time is the greatest enemy. Human minds are incredible at understanding and theorizing, if given enough time. But when we are young and lacking information, pressure and trust of authority create a time-restriction that disallows us from forming truly "soft" or flexible models. Thus we are sort of taught to "forget" about a problem once it is solved. The problem itself becomes a task to be finished (for brownie points) rather than an actual problem with its own unique set of rewards. Not every problem rewards equally and even some problems may be detrimental to solve, but the self-education that our education system requires of our minds to succeed in the system makes it such that we regard problems as something we need to solve as proof to authority that we may get some free time.
Heh I get this a lot in my daily sysadmin/developer duties. When you need to turn on a knob somewhere you don't just wanna know that you have to turn it on, you want to know why you need to turn it on and follow the chain up until you get to facts you already know. But it's not always possible to get that far, there's too many layers of abstraction, the source code is not available or you just don't have the time.
I can draw a parallel to this in software development. Some product features that require development "from scratch", where you can get down to the original code and logic - this is where "taking time to think" really pays off.
But when you are basically composing a final product from components, libraries and features - this is where figuring something out may take really long time and a lot of effort. It today's world many libraries are open source, so you actually can get to the bottom of many issues. But the time and effort cost of that is almost never acceptable.
My conclusion is - if you are a "slow" thinker, prefer getting to the bottom and figuring stuff out - try and choose the "fundamental" type of work. Where you are "done is better than perfect" kinda person - you'll thrive in the upper layers of development stack where shipping stuff out is of utmost importance. Focus on your strengths.
Swimming in uncertainty and (conditionally) trusting abstractions is one of the skills I have to teach my interns. I love their desire to understand and I try to be careful not to kill it. But at the same time, software engineering in the real world means contributing to a picture that is larger than will fit in your head.
Agree on the aspect of time. As we have progressed, we now have many levels of abstraction that it is hard to think deeply about the problem. Almost à la like a code that has grown too deep to understand every "bit" of it.
Moreover, I think people now work in teams rather than one individual thinking about the system holistically.
It’s possible. It just takes time; very few people are in a position to devote that time.
This gives an advantage I haven’t seen discussed: when you put in the time, you make connections no one else thought of. It happens time and again, and it’s a clear pattern at this point.
It takes months of daily study, often tedious, with no clear benefit. But the benefits sometimes come. (I wrote “usually” rather than “sometimes,” but that’s not really true. The usual result is that you go to sleep more confused than you started. It’s not till much, much later that the connections even seem relevant.)
i agree. I think this is why I find companies like tesla and spacex exciting. They seem to have set up incentive structures that encourage _both_ quick execution and innovation (which requires deep understanding). One thing he's said that really struck me is that it's _really_ difficult to produce innovation if you tie punishment to failure. People tend to be conservative if they are punished / think they will be punished harshly for trying and failing. But if you want to innovate, failure has to be an acceptable outcome
hopefully we see more companies go in this direction
well that's where the whole mantra of "move fast and break things" comes from.
Putting it out there and failing also accelerates you faster to the right answers. If you release it today, it'll take 6 more months of iteration to really get it right. Or maybe you spend an extra 2 years of development to get it "right", but then once you release, you'll still have to spend 3 more months of iteration anyways to get it right.
I would say the IAS is one, as are other pure research institutes like Perimeter.
I would also say that a deep understanding isn’t usually necessary to make progress in many fields of human endeavor. Fields like engineering work on the basis of empiricism — theoretical understanding usually comes later. I could be all wrong here but my gut feeling is that the majority of great breakthroughs in engineering have come through tinkering and luck rather than a principled application of science — the latter is used more for refining the execution.
Even in recent times, neural network models have been shown to work without any deep understanding apart from the basics. It’s only recently that a lot of new theory have come out.
Deep understanding is a worthy goal in order to deconstruct things to learn how to make them fundamentally better or to build a foundation for further progress.
But as humans, we do very well surviving in a world that we largely do not deeply understand for the most part — I’d argue we are able to do so through heuristics (see Gigerenzer). We definitely should not let the lack of deep understanding prevent us from taking the initiative to do things.
The points about education really resonate with me. In my engineering undergrad, it was frustrating to see in our applied math courses that the mechanical plugging and chugging of equations was the approach that most of my peers took in their studies. They got better grades than me. I wanted to understand concepts more deeply, but there was no time, and the tests rewarded those who could simply go through the motions of applying formulae to problems.
I'm a first-year engineering undergrad, last Friday I finished the last Calculus 3 exam on the semester.
I spent 2 days studying with friends just to see they would blindly memorize formulas with no regard whatsoever for what they were actually doing. "I'm not the understanding type", they said unironically.
Do you think this happens more often in engineering than other disciplines? Some people believe that applied science or mathematics means that just learning formulas is enough
I was one of those people and I regret it. However, the course load was just so great that I'm not sure how I'd done it the "slow" way. I wish I had the time to go back to school and do it right.
> 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.
Readit News
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 theorems, I'd memorised some proofs, and I could occasionally manipulate a problem to get an answer. I think about half of the cohort was in a similar position. But it was clear that there were others in a completely different league.
When we were studying for our first year exams, I was struggling to remember the proof of a specific theorem (it felt quite long). A friend was trying to help me learn it, and he asked me what "picture" I had in my head for the theorem. I didn't have any pictures in my head for anything.
It turned out that a simple drawing could capture the entire statement of the theorem, and from that drawing, the proof was trivial to derive. It was long-ish to write out in words, sure, but the underlying concept was really simple. This blew my mind — I realised I didn't have a clue what we'd been studying the whole year.
The worrying thing is that I actually thought I understood that stuff. Before that incident, I didn't know what it feels like to actually understand something, and I didn't have an appreciation for the layers of depth even within that. I suspect lots of people go through the entire education system like this.
Excellent point.
1. A similar example: Feynman diagrams.
2. Another: Venn diagrams.
3. Longer example: On Navy nuclear-powered aircraft carriers, the officer of the deck underway (OOD) must have at least a basic understanding of how the engineering plant works. It's second nature for nuclear-trained OODs, of course, but non-nukes could sometimes have trouble. Back in the day, it turned out that an effective way to help non-nukes learn what they needed to know was to have them: (A) memorize a really-simple block diagram of the reactor and steam system, and also (B) memorize a chant, of modest length, that summarized how things worked. During slow periods while standing OOD watch, I'd make a non-nuke OOD trainee draw the diagram from memory; then I'd quiz him with "what if ..." questions (back then it was always "him"). If he got hung up on a question, I'd tell him, "chant the chant." That usually helped him figure out the answer in short order.
(U.S. submarines don't have that problem, AFAIK, because pretty much every officer who will stand OOD watches is nuclear-trained.)
+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 them and says you need to master each step on the track before moving onto the next one. this is poisonous and borderline evil, and I've encountered many people who are crippled for life by it. a lot of people never pursue things they're really interested in and could become extremely passionate about because school has convinced them they need to stack up prerequisite knowledge before they're even allowed to touch it
When reading Celine, one understand that children are natural born learner, and there is no effort needed to make them learn stuff. Our school model is industrial production of objects. Thinking human machines. We are way more than that. Sadly Pink Floyd description of the school still echo to our modern school. Some peoples don't feel that way about school. I don't really know why. Maybe they never imagined how better it could have been, so they found it great.
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.
The way you phrased this reminded me of A Mathematician's Lament by Paul Lockhart[0]
[0] https://www.maa.org/external_archive/devlin/LockhartsLament....
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 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.
This is the theorem I was talking about: https://i.imgur.com/1xEH51Z.png (taken from https://taimur.me/posts/thinking-at-the-right-level-of-abstr... which touches on a similar topic to your post)
[0] https://en.wikipedia.org/wiki/Lagrange%27s_theorem
[1] https://i.stack.imgur.com/w2GfA.png
So now I try to “do” instead of memorize and it makes all the difference in the world.
Even I can’t learn new programming languages unless I make a real project with them. That’s why I don’t believe in things like code katas or reading programming books cover to cover. It’s a bad representative for using a tool in real life to learn its principles, techniques and methods.
But it's good news that you were able to understand once he drew the picture for you. So there is an effective way to teach that theorem - If only the professor knew it.
Creating really unique pneumonics works similar to how you can recall when something happened out of the ordinary or have a "sixth" sense where you subconsciously can sense something is off in an otherwise ordinary situation or encounter. If I had to guess, this might be an evolved human survival mechanism that helped our ancestors detect and avoid dangerous/deadly situation.
I wonder what other evolutionary mechanism we can hack to encode concepts more deeply. I know spaced repetition is the mainstream approach via flashcards/ankii type apps. I think pneumonics blended with spaced repetition is the most practical methodology. Thoughts?
Couldn't agree more. Some of the best CS books I've read have been ones with a tonne of illustrations, like the Head First series.
Humans are wired to enjoy and remember stories. So much education is done narrativelessly and as such neglects to harness one of our most potent capabilities.
Did you do a lot of exercises in school? They usually help build the intuitive part, the aha moment, that comes through repetition.
We did do a lot of exercises in school, but they mostly just tested whether you can reliably apply a method that you were taught, rather than testing understanding.
I'm not into showing off ranking or pedigree but I do genuinely believe that the higher the pedigree your school the more likely the exams will be harder and require total understanding and even creativity over rote memorization.
The reason is because memorization is trivial. Students able to get into any top school will likely all easily achieve full score on an ordinary tests. The professors at top schools need to make these tests brutally hard in order to produce a bell curve.
I literally had one new professor at my school actually give a mid term that was what would ordinarily be called fair in any other school or college... but the entire class ended up getting nearly full score.
He realized his mistake and the final was way, way harder.
In Asia it is common for schools to solely grade base on multiple choice exams which promote cramming and memorization. But in the US I would be shocked if a program that only operated that way was accredited. I would argue students who simply memorize things are trying to hack their way to an unearned grade. The goal of higher education is supposed to be to reach higher levels of Bloom's taxonomy, beyond the rote stuff done in primary and secondary school.
This is no longer how “high pedigree” schools function in the US. There is too much blowback from helicopter parents and the kids who haven’t ever had to deal with critical feedback before.
Lookup “grade inflation” if you want to be horrified by the quality decline of “pedigree education”.
> 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 around you will urge you to "stop thinking just do the task" or "do the obvious thing" as they lose patience with you. If you don't resist them you'll end up moving forward despite feeling confused, sometimes completely. You'll then end up pissing people off when you execute too slowly or fail (in their eyes, intentionally).
> This is a habit. It’s easy to pick up.
Not if the people around you are exhausted by you.
More so if you're a college student that can focus 100% on your academic life/studies, unburdened by things like work.
EVEN more so if you have great mentors, professors, etc. that can guide you to the right place.
I'm not saying that one CAN NOT do the things above, if you're a busy student with work on the side, and very limited resources as far as mentors or professors go...but I do think that those lucky enough to find themselves in the right positions, are more likely to mature - and quicker.
(And it was no surprise to see that the author is a PPE grad from Oxford)
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Often, trying (and potentially failing) to do a task is the best way to learn about it. The key is to be very explicit about what parts of that task you actually understand and which ones you're pulling out of your ass.
This is especially true when creating software. It's super rare to have requirements that are concrete and detailed enough to form a comprehensive understanding of the final design before you start developing it. Instead there are usually parts that have clarity and others that are fuzzy. If you can enumerate those and keep them separate you can often leverage the parts you understand to make progress on those you do not. Writing placeholder/obviously-terrible code to stand in for the unknown parts just so that you can spin up a running system is a great way to do that. Along the way you'll see what patterns emerge, where you hit walls, etc, which is not always easy to imagine with raw abstract thought. And having "working" software that you can play with is a great way to find edge cases and otherwise make progress on those unknowns. Once you've gained a more complete understanding you can replace the placeholder junk with well-designed/actually-thought-out modules.
(Another obvious reason to do this is if your company will literally go out of business if you wait until you have a perfect understanding to launch a product/service, but I think most people here get that.)
I'm not sure how much this generalizes, but it also works well for me when writing. I usually start with a vague understanding of an idea I hope to communicate, then jot down disjointed sentences to capture parts of it. As I do so it gradually becomes clear how things are connected, where my reasoning is muddy, what I thought I knew but can't express so probably don't, etc, and I can use this gained understanding to iteratively rewrite and reshape my message until it becomes something coherent. Sometimes, anyway; other times I don't end up sending/publishing it at all because along the way I learned that the thing I was hoping to communicate was based on a faulty assumption or is not as straightforward as I thought it was. Which is great, because either way I've learned something.
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I guess I'd say it differently: "keep thinking and do the task".
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A common unspoken theme in the post is, the art of learning is knowing "what" to care about and "when".
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.
Example: I forget the name of the principle, but in mathematics the statement "P implies Q" is considered true if P can never be true. As an example, let P be "George Washington was a woman" and Q be "Queen Elizabeth is a man". Then the statement "If GW was a woman, then QE is a man" is considered to be a true statement.
I have a friend who refuses to accept that such a statement should be considered "true". And he has put off studying real analysis until he can learn enough logic theory to convince himself on the validity of accepting such statements as true. I do not think he'll ever get to study real analysis, because he is full of "No! I need to understand this really really well before proceeding!" statements.
It's a fine approach if you have an infinite amount of time.
The other issue, as another commenter pointed out: It's very difficult to measure progress in thought. The mind is great at fooling itself, and not until you try to solve real problems (or discuss them with others) will you expose most of the gaps in your mind. The same person in the above anecdote does suffer from this. He definitely puts in effort to learn a lot (and has succeeded), but there are always more things to learn, and he moves on to the next topic before really applying what he has learned. As someone who talks to him often, it's really hard to tell if he understands. He is the classic case of "I'm sure I can solve problems when I need to, with a bit of review".
At the other extreme, of course, are people who are not really that motivated to understand. They are satisfied if they get the answer at the back of the book. You won't get far with just that.
> If you don't resist them you'll end up moving forward despite feeling confused, sometimes completely. You'll then end up pissing people off when you execute too slowly or fail (in their eyes, intentionally).
This sounds more like an issue at work, and your experience is fairly universal - most jobs I've worked at have it. In my experience, understanding things well is sadly not valued on the job. They want you to "execute", and want you to minimize the time you spend learning. And of course, they would rather hire someone else instead of ensuring your proper learning/training.
You can keep multiplying your equation by x^0 and union-ing your set with the empty set and or-ing your proposition with "If eggs are diamonds then fish can talk" all the livelong day, but you never gain any more information.
> I have a friend who refuses to accept that such a statement should be considered "true".
This is a weird thing to refuse to accept, since it is an ordinary part of vernacular English, a common way of dismissing an assertion as false.
"If that guy graduated high school, I'm the King of England."
I have to agree with your friend :-)
In general, there are a bunch of starting conditions necessary for this kind of approach to glean value.
Great post.
But when you are basically composing a final product from components, libraries and features - this is where figuring something out may take really long time and a lot of effort. It today's world many libraries are open source, so you actually can get to the bottom of many issues. But the time and effort cost of that is almost never acceptable.
My conclusion is - if you are a "slow" thinker, prefer getting to the bottom and figuring stuff out - try and choose the "fundamental" type of work. Where you are "done is better than perfect" kinda person - you'll thrive in the upper layers of development stack where shipping stuff out is of utmost importance. Focus on your strengths.
This gives an advantage I haven’t seen discussed: when you put in the time, you make connections no one else thought of. It happens time and again, and it’s a clear pattern at this point.
It takes months of daily study, often tedious, with no clear benefit. But the benefits sometimes come. (I wrote “usually” rather than “sometimes,” but that’s not really true. The usual result is that you go to sleep more confused than you started. It’s not till much, much later that the connections even seem relevant.)
1. Industry cares more about concrete results, quick execution, and bias for action.
2. Academia cares more about positive results, quantity of published papers, and small achievable experiments over big experiments that might fail.
Where are the institutions that care about deep understanding?
hopefully we see more companies go in this direction
Putting it out there and failing also accelerates you faster to the right answers. If you release it today, it'll take 6 more months of iteration to really get it right. Or maybe you spend an extra 2 years of development to get it "right", but then once you release, you'll still have to spend 3 more months of iteration anyways to get it right.
I would also say that a deep understanding isn’t usually necessary to make progress in many fields of human endeavor. Fields like engineering work on the basis of empiricism — theoretical understanding usually comes later. I could be all wrong here but my gut feeling is that the majority of great breakthroughs in engineering have come through tinkering and luck rather than a principled application of science — the latter is used more for refining the execution.
Even in recent times, neural network models have been shown to work without any deep understanding apart from the basics. It’s only recently that a lot of new theory have come out.
Deep understanding is a worthy goal in order to deconstruct things to learn how to make them fundamentally better or to build a foundation for further progress.
But as humans, we do very well surviving in a world that we largely do not deeply understand for the most part — I’d argue we are able to do so through heuristics (see Gigerenzer). We definitely should not let the lack of deep understanding prevent us from taking the initiative to do things.
I spent 2 days studying with friends just to see they would blindly memorize formulas with no regard whatsoever for what they were actually doing. "I'm not the understanding type", they said unironically.
Do you think this happens more often in engineering than other disciplines? Some people believe that applied science or mathematics means that just learning formulas is enough
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.