7 ms·

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 meth

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 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.

ReactiveJelly 6y ago
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.
nqureshi 6y ago
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.)
- refrigerator 6y ago
  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" This is the theorem I was talking about: https://i.imgur.com/1xEH51Z.png https://i.imgur.com/1xEH51Z.png (taken from https://taimur.me/posts/thinking-at-the-right-level-of-abstraction/ https://taimur.me/posts/thinking-at-the-right-level-of-abstr... which touches on a similar topic to your post)
- Chris_Newton 6y ago
  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 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.
  - nqureshi 6y ago
    Awesome example, thank you!
  - ericjang 6y ago
grugagag 6y ago
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.
- refrigerator 6y ago
  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.
johnchristopher 6y ago
Did that realization at the eve of exams bring up a bout of anxiety ? (edit: I would have freaked out)
- refrigerator 6y ago
  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.
alicemaz 6y ago
>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
- knzhou 6y ago
  > 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.
  - pc86 6y ago
    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.
    - alicemaz 6y ago
      I'm a believer in individual learning styles, and hard as it is for me to understand personally, I do recognize that lots of people may learn best by lecture and drill. I certainly don't think bad of them for it. but many more people are not suited for this, and forcing them through the education system as it exists does them great harm, and I empathize strongly with them
dctoedt 6y ago
> 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.)
leafboi 6y ago
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.
- kortilla 6y ago
  > 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”.
  - leafboi 6y ago
    You got a good point. My school is listed as a school that has grade inflation but anecdotally I feel my engineering department was pretty brutal. Average GPA was below 3 when I was there. It could be my personal situation was not representative of the norm.
- unishark 6y ago
  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.
sizzle 6y ago
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?
eric4smith 6y ago
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.
ignoramous 6y ago
> 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. 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.
- ftio 6y ago
  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.
jairofloress 6y ago
I always remember the exams frustration of remembering the exact page of some graph but being unable to get it back. I hated it!

ReactiveJelly 6y ago

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.

nqureshi 6y ago

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.)

refrigerator 6y ago
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" This is the theorem I was talking about: https://i.imgur.com/1xEH51Z.png https://i.imgur.com/1xEH51Z.png (taken from https://taimur.me/posts/thinking-at-the-right-level-of-abstraction/ https://taimur.me/posts/thinking-at-the-right-level-of-abstr... which touches on a similar topic to your post)
Chris_Newton 6y ago
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 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.
- nqureshi 6y ago
  Awesome example, thank you!
- ericjang 6y ago

grugagag 6y ago

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.

refrigerator 6y ago
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.

johnchristopher 6y ago

Did that realization at the eve of exams bring up a bout of anxiety ? (edit: I would have freaked out)

refrigerator 6y ago
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.

alicemaz 6y ago

>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

knzhou 6y ago
> 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.
- pc86 6y ago
  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.
  - alicemaz 6y ago
    I'm a believer in individual learning styles, and hard as it is for me to understand personally, I do recognize that lots of people may learn best by lecture and drill. I certainly don't think bad of them for it. but many more people are not suited for this, and forcing them through the education system as it exists does them great harm, and I empathize strongly with them

dctoedt 6y ago

> 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.)

leafboi 6y ago

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.

kortilla 6y ago
> 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”.
- leafboi 6y ago
  You got a good point. My school is listed as a school that has grade inflation but anecdotally I feel my engineering department was pretty brutal. Average GPA was below 3 when I was there. It could be my personal situation was not representative of the norm.
unishark 6y ago
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.

sizzle 6y ago

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?

eric4smith 6y ago

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.

ignoramous 6y ago

> 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. 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.

ftio 6y ago
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.

jairofloress 6y ago

I always remember the exams frustration of remembering the exact page of some graph but being unable to get it back. I hated it!