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In undergrad I took an abstract algebra class. It was very difficult and one of the things the teacher did was have us memorize proofs. In fact, all of his test

In undergrad I took an abstract algebra class. It was very difficult and one of the things the teacher did was have us memorize proofs. In fact, all of his tests were the same format: reproduce a well-known proof from memory, and then complete a novel proof. At first I was aghast at this rote memorization - I maybe even found it offensive. But an amazing thing happened - I realized that it was impossible to memorize a proof without understanding it! Moreover, producing the novel proofs required the same kinds of "components" and now because they were "installed" in my brain I could use them more intuitively. (Looking back I'd say it enabled an efficient search of a tree of sequences of steps). Memorization is not a panacea. I never found memorizing l33t code problems to be edifying. I think it's because those kinds of tight, self-referential, clever programs are far removed from the activity of writing applications. Most working programmers do not run into a novel algorithm problem but once or twice a career. Application programming has more the flavor of a human-mediated graph-traversal, where the human has access to a node's local state and they improvise movement and mutation using only that local state plus some rapidly decaying stack. That is, there is no well-defined sequence for any given real-world problem, only heuristics.

MattPalmer1086 1y ago
Hmmm... It's the other way around for me. I find it hard to memorise things I don't actually understand. I remember being given a proof of why RSA encryption is secure. All the other students just regurgitated it. It made superficial sense I guess. However, I could not understand the proof and felt quite stupid. Eventually I went to my professor for help. He admitted the proof he had given was incomplete (and showed me why it still worked). He also said he hadn't expected anyone to notice it wasn't a complete proof.
- benterix 1y ago
  >> I realized that it was impossible to memorize a proof without understanding it! > I find it hard to memorise things I don't actually understand. Isn't it the parent's point?
  - MattPalmer1086 1y ago
    Yep, I misread the parent post.
- jghn 1y ago
  > Hmmm... It's the other way around for me. I find it hard to memorise things I don't actually understand. I think you two are agreeing. GP said that they found they couldn't memorize something until they actually understood it
  - MattPalmer1086 1y ago
    You're correct, I read it the wrong way!
    - blackoil 1y ago
      Hope now you'll remember it :P
- widforss 1y ago
  But, is it proven that RSA is secure? Wouldn't that also prove P != NP?
  - MattPalmer1086 1y ago
    Haha, well it does depend on the assumption that integer factorisation is hard. Although I'm not sure that being able to do it implies P = NP.
- pigeons 1y ago
  > I remember being given a proof of why RSA encryption is secure With what assumptions?
  - MattPalmer1086 1y ago
    Mostly just integer factorisation of large numbers is hard. There are some other things you have to worry about practically, e.g Coppersmith's attack, and padding schemes (although that wasn't part of the proof I was given)
WhitneyLand 1y ago
Your comment about memorizing as part of understanding makes a lot of sense to me, especially as one possible technique to get get unstuck in grasping a concept. If it doesn’t work for you on l33t code problems, what techniques are you finding more effective in that case?
- jdmichal 1y ago
  I was part of an ACM programming team in college. We would review classes of problems based on the type of solution necessary, and learn those techniques for solving them. We were permitted a notebook, and ours was full of the general outline of each of these classes and techniques. Along with specific examples of the more common algorithms we might encounter. As a concrete example, there is a class of problems that are well served by dynamic programming. So we would review specific examples like Dijkstra's algorithm for shortest path. Or Wagner–Fischer algorithm for Levenshtein-style string editing. But we would also learn, often via these concrete examples, of how to classify and structure a problem into a dynamic programming solution. I have no idea if this is what is meant by "l33t code solutions", but I thought it would be a helpful response anyway. But the bottom line is that these are not common in industry, because hard computer science is not necessary for typical business problems. The same way you don't require material sciences advancements to build a typical house. Instead it flows the other way, where advancements in materials sciences will trickle down to changing what the typical house build looks like.
- marcosdumay 1y ago
  What I understand from the GP is that memorizing l33t code won't help you learn anything useful. Not that understanding the solutions won't help you memorize them.
- simpaticoder 1y ago
  >If it doesn’t work for you on l33t code problems, what techniques are you finding more effective in that case? Memorization of l33t code DOES work well as prep for l33t code tests. I just don't think l33t code has much to do with application programming. I've long felt that "computer science" is physics for computers, low on the abstraction ladder, and there are missing labels for the higher complexity subjects built on it. Imagine if all physical sciences were called "physics" and so in order to get a job as a biologist you should expect to be asked questions about the Schroedinger equation and the standard model. We desperately need "application engineering" to be a distinct subject taught at the university level.
bitexploder 1y ago
Memorizing is a super power / skill. I work in a ridiculously complex environment and have to learn and know so much. Memorizing and spaced repetition are like little islands my brain can start building bridges between. I used to think memorizing was anti-first principles, but it is just good. Our brains can memorize so much if we make them. And then we can connect and pattern matching using higher order thinking.
- lvspiff 1y ago
  Recognizing the patterns and applying patterned solutions is where I see success in my niche of healthcare interoperability. So much of my time is spent watching people do things,process and how they use data. It's amazing how much people remember to do their job, but me coming in and be able to bridge the doctor and the lab to share data easier is like Im an alchemist. It's really not a problem I've been able to see ai solve without suggesting solutions that are too simple or too costly and in that goldilocks zone everyone will be happy with
- simpaticoder 1y ago
  What's even better about memorization is that you have an objective method to test your own understanding. It is so easy to believe you understand something when you don't! But, at least with math, I think if you can reproduce the proof from memory you can be very confident that you aren't deluding yourself.
- supriyo-biswas 1y ago
  There's this [1] which is one of my favorite articles on that topic; definitely worth a read. [1] https://www.pearlleff.com/in-praise-of-memorization https://www.pearlleff.com/in-praise-of-memorization
- hosh 1y ago
  In education, I have heard it called “fluency”.
trod1234 1y ago
Mathematics pedagogy today is in a pretty sorrowful state due to bad actors and willful blindness at all levels that require public trust. A dominant majority in public schools starting late 1970s seems to follow the "Lying to Children" approach which is often mistakenly recognized as by-rote teaching but are based in Paulo Freire's works that are in turn based on Mao's torture discoveries from the 1950s. This approach contrary to classical approaches leverages torturous process which seems to be purposefully built to fracture and weed out the intelligent individual from useful fields, imposing sufficient thresholds of stress to impose PTSD or psychosis, selecting for and filtering in favor of those who can flexibly/willfully blind/corrupt themselves. Such sequences include Algebra->Geometry->Trigonometry where gimmicks in undisclosed changes to grading cause circular trauma loops with the abandonment of Math-dependent careers thereafter, similar structures are also found in Uni, for Economics, Business, and Physics which utilize similar fail-scenarios burning bridges where you can't go back when the failure lagged from the first sequence, and you passed the second unrelated sequence. No help occurs, inducing confusion and frustration to PTSD levels, before the teacher offers the Alice in Wonderland Technique, "If you aren't able to do these things, perhaps you shouldn't go into a field that uses it". (ref Kubark Report, Declassified CIA Manual) Have you been able to discern whether these "patterns" as you've called them aren't just the practical reversion to the classical approach (Trivium/Quadrivium)? Also known as the first-principles approach after all the filtering has been done. To compare: Classical approaches start with nothing but a useful real system and observations which don't entrench false assumptions as truth, which are then reduced to components and relationships to form a model. The model is then checked for accuracy against current data to separate truth from false in those relationships/assertions in an iterative process with the end goal being to predict future events in similar systems accurately. The approach uses both a priori and a posteriori components to reasoning. Lying to Children reverses and bastardizes this process. It starts with a single useless system which contains equal parts true and false principles (as misleading assumptions) which are tested and must be learned to competency (growing those neurons close together). Upon the next iteration one must unlearn the false parts while relearning the true parts (but we can't really unlearn, we can only strengthen or weaken) which in turn creates inconsistent mental states imposing stress (torture). This is repeated in an ongoing basis often circular in nature (structuring), and leveraging psychological blindspots (clustering), with several purposefully structured failings (elements) to gatekeep math through torturous process which is the basis for science and other risky subject matter. As the student progresses towards mastery (gnosis), the systems become increasingly more useful. One must repeatedly struggle in their sessions to learn, with the basis being if you aren't struggling you aren't learning. This mostly uses a faux a priori reasoning without properties of metaphysical objectivity (tied to objective measure, at least not until the very end). If you don't recognize this, an example would be the electrical water pipe pressure analogy. Diffusion of charge in-like materials, with Intensity (Current) towards the outermost layer was the first-principled approach pre-1978 (I=V/R). The Water Analogy fails when the naive student tries to relate the behavior to pressure equations that ends up being contradictory at points in the system in a number of places introducing stumbling blocks that must be unlearned. Torture being the purposefully directed imposition of psychological stress beyond a individuals capacity to cope towards physiological stages of heightened suggestability and mental breakdown (where rational thought is reduced or non-existent in the intelligent). It is often recognized by its characteristic subgroups of Elements (cognitive dissonance, a lack of agency to remove oneself and coercion/compulsion with real or perceived loss or the threat thereof), Structuring (circular patterns of strictness followed by leniency in a loop, fractionation), and Clustering (psychological blindspots).
raincole 1y ago
During my elementary school years, there was a teacher who told me that I didn't need to memorize it as long as I understand them. I taught he was the coolest guy ever. Only when I got late twenties, I realized how wrong he was. Memorization and understanding go hand in hand, but if one of them has to come first than it's memorization. He probably said that because that was what kids (who were forced to do rote memorization) wanted to hear.
- HankStallone 1y ago
  Probably. I hated memorization when I was a student too, because it was boring. But as soon as I did some teaching, my attitude changed to, "Just memorize it, it'll make your life so much easier." It's rough watching kids try to multiply when they don't have their times tables memorized, or translate a language when they haven't memorized the vocabulary words in the lesson so they have to look up each one.
  - skydhash 1y ago
    There's things that you need to know (2*2 = 4) and there are things that you need to understand (multiplication rules). Both can happen with practice, but they're not that related. Memorization is more like a shortcut. You don't need to go through the problem solving process to know the result. But with understanding, you master the heuristic factors needed to know when to take the shortcut and when to go through the problem solving route. The Dreyfus Skill Model [0] is a good explanation. Novice typically have to memorize, then as they master the subject, their decision making becomes more heuristic based. LLMs don't do well with heuristics, and by the times you've nailed down all the problems data, you could have been done. What they excels at is memorization, but all the formulaic stuff have been extracted into frameworks and libraries for the most popular languages. [0]: https://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisition#The_Dreyfus_Skill_Model https://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisi...
    - mason55 1y ago
      I think the problem is that in spots where the concepts build on one another, you need to memorize the lower level concepts or else it'll be too hard to make progress on the higher level concepts. If you're trying to expand polynomials and you constantly have to re-derive multiplication from first principles, you're never going to make any progress on expanding polynomials.
stonemetal12 1y ago
>Memorization is not a panacea. It is What you memorize that is important, you can't have a good discussion about a topic if you don't have the facts and logic of the topic in memory. On the other hand using memory to paper over bad design instead of simplifying or properly modularizing it, leads to that 'the worst code I have seen is code I wrote six months ago' feeling.
tshaddox 1y ago
> But an amazing thing happened - I realized that it was impossible to memorize a proof without understanding it! This may be true of mathematical proofs, but it surely must not be true in general. Memorizing long strings of digits of pi probably isn’t much easier if you understand geometry. Memorizing famous speeches probably isn’t much easier if you understand the historical context.
- gundmc 1y ago
  > Memorizing famous speeches probably isn’t much easier if you understand the historical context. Not commenting on the merits of critical thinking vs memorization either way, but I think it would be meaningfully easier to memorize famous speeches if you understand the historical context.
  - codr7 1y ago
    Right, isn't building up a (imaginary) context how people memorize pi?
  - tshaddox 1y ago
    For memorizing a speech word-for-word, I don't think so. Knowing the years of the signing of the Declaration of Independence and the Gettysburg Address aren't gonna help you nail the exact wording of the first sentence.
taeric 1y ago
I would wager some folks can memorize without understanding? I do think memorization is underrated, though. There is also something to the practice of reproducing something. I always took this as a form of "machine learning" for us. Just as you get better at juggling by actually juggling, you get better at thinking about math by thinking about math.
- lazide 1y ago
  Rote memorization is essentially that, yes.
Aurornis 1y ago
My controversial education hot take: Pointless rote memorization is bad and frustrating, but early education could use more directed memorization. As you discovered: A properly structured memorization of carefully selected real world material forces you to come up with tricks and techniques to remember things. With structured information (proofs in your case) you start learning that the most efficient way to memorize is to understand, which then reduces the memorization problem into one of categorizing the proof and understanding the logical steps to get from one step to another. In doing so, you are forced to learn and understand the material. Another controversial take (for HN, anyway) is that this is what happens when programmers study LeetCode. There’s a meme that the way to interview prep is to “memorize LeetCode”. You can tell who hasn’t done much LeetCode interviewing if they think memorizing a lot of problems is a viable way to pass interviews. People who attempt this discover that there are far too many questions to memorize and the best jobs have already written their own questions that aren’t out of LeetCode. Even if you do get a direct LeetCode problem in an interview, a good interview will expect you to explain your logic, describe how you arrived at the solution, and might introduce a change if they suspect you’re regurgitating memorized answers. Instead, the strategy that actually works is to learn the categories of LeetCode style questions, understand the much smaller number of algorithms, and learn how to apply them to new problems. It’s far easier to memorize the dozen or so patterns used in LeetCode problems (binary search, two pointers, greedy, backtracking, and so on) and then learn how to apply those. By practicing you’re not memorizing the specific problems, you’re teaching yourself how to apply algorithms. Side note: I’m not advocating for or against LeetCode, I’m trying to explain a viable strategy for today’s interview format.
- bee_rider 1y ago
  Memorization of, like, multiplication tables gives us a poor view of the more interesting type of memorization. Remembering types of problems we’ve seen. Remembering landmarks and paths, vs just remembering what’s in every cell of a big grid. I still don’t like leetcode, though.
  - Aurornis 1y ago
    > Memorization of, like, multiplication tables gives us a poor view of the more interesting type of memorization. Memorizing multiplication tables is the first place many children encounter this strategy: The teacher shows you that you could try to memorize all of the combinations, or you could start learning some of the patterns and techniques. When multiplying by 5 the answer will end in 0 or 5. When multiplying by 2 the answer will be an even number, and so on.
dunham 1y ago
Fortunately that was not my experience in abstract algebra. The tests and homework were novel proofs that we hadn't seen in class. It was one of my favorite classes / subjects. Someone did tell me in college that they did the memorization thing in German Universities. Code-wise, I spent a lot of time in college reading other people's code. But no memorization. I remember David Betz advsys, Tim Budd's "Little Smalltalk", and Matt Dillon's "DME Editor" and C compiler.
- jdougan 1y ago
  Another advsys enjoyer! Did you ever write a game with it?
glitchc 1y ago
Indeed, not just math. Biology requires immense amounts of memorization. Nature is littered with exceptions.
johntarter 1y ago
Interesting I had the same problem and suffered in grades back in school simply because I couldn't memorize much without understanding. However, I seemed to be the only one because every single other student, including those with top grades, were happy to memorize and regurgitate. I wonder how they're doing now.
ghelmer 1y ago
I find it's helpful to have context to frame what I'm memorizing to help me understand the value.
eithed 1y ago
Depends on the subject - I can remember multiple subjects where the teacher would give you a formula to memorise without explaining why or where it came from. You had to take it as an axiom. The teachers also didn't say - hey, if you want to know why did we arrive to this, have a read here, no, it was just given. Ofc you could also say that's for the student to find out, but I've had other things on my mind
ian-g 1y ago
It's funny, because I had the exact opposite experience with abstract algebra. The professor explained things, we did proofs in class, we had problem sets, and then he gave us open-book semi-open-professor take-home exams that took us most of a week to do. Proof classes were mostly fine. Boring, sometimes ridiculously shit[0], but mostly fine. Being told we have a week for this exam that will kick our ass was significantly better for synthesizing things we'd learned. I used the proofs we had. I used sections of the textbook we hadn't covered. I traded some points on the exam for hints. And it was significantly more engaging than any other class' exams. [0] Coming up with novel things to prove that don't require some unrelated leap of intuition that only one student gets is really hard to do. Damn you Dr. B, needing to figure out that you have to define a third equation h(x) as (f(x) - g(x))/(f(x) + g(x)) as the first step of a proof isn't reasonable in a 60 minute exam.
__alexs 1y ago
Is it the memorisation that had the desired effect or the having to come up with the novel proofs? Many schools seem to do the memorising part, but not the creating part.
bwfan123 1y ago
My abstract algebra class had it exactly backwards. It started with a lot of needless formalism culminating in galois theory. This was boring to most students as they had no clue why the formalism was invented in the first place. Instead, I wished it showed how the sausage was actually made in the original writings of galois [1]. This would have been far more interesting to students, as it showed the struggles that went into making the product - not to mention the colorful personality of the founder. The history of how concepts were invented for the problems faced is far more motivating to students to build a mental model than canned capsules of knowledge. [1] https://www.ams.org/notices/201207/rtx120700912p.pdf https://www.ams.org/notices/201207/rtx120700912p.pdf
- inetknght 1y ago
  > This was boring to most students as they had no clue why the formalism was invented in the first place. > The history of how concepts were invented for the problems faced is far more motivating to students to build a mental model than canned capsules of knowledge. That's something I really like about 3blue1brown, and he says it straight up [0]: > My goal is for you to come away feeling like you could have invented calculus yourself. That is, cover all those core ideas, but in a way that makes clear where they actually come from, and what they really mean, using an all-around visual approach. [0]: https://www.youtube.com/watch?v=WUvTyaaNkzM https://www.youtube.com/watch?v=WUvTyaaNkzM
thoughtlede 1y ago
memorization + application = comprehension. Rinse and repeat. Whether leet code or anything else.

MattPalmer1086 1y ago

Hmmm... It's the other way around for me. I find it hard to memorise things I don't actually understand. I remember being given a proof of why RSA encryption is secure. All the other students just regurgitated it. It made superficial sense I guess. However, I could not understand the proof and felt quite stupid. Eventually I went to my professor for help. He admitted the proof he had given was incomplete (and showed me why it still worked). He also said he hadn't expected anyone to notice it wasn't a complete proof.

benterix 1y ago
>> I realized that it was impossible to memorize a proof without understanding it! > I find it hard to memorise things I don't actually understand. Isn't it the parent's point?
- MattPalmer1086 1y ago
  Yep, I misread the parent post.
jghn 1y ago
> Hmmm... It's the other way around for me. I find it hard to memorise things I don't actually understand. I think you two are agreeing. GP said that they found they couldn't memorize something until they actually understood it
- MattPalmer1086 1y ago
  You're correct, I read it the wrong way!
  - blackoil 1y ago
    Hope now you'll remember it :P
widforss 1y ago
But, is it proven that RSA is secure? Wouldn't that also prove P != NP?
- MattPalmer1086 1y ago
  Haha, well it does depend on the assumption that integer factorisation is hard. Although I'm not sure that being able to do it implies P = NP.
pigeons 1y ago
> I remember being given a proof of why RSA encryption is secure With what assumptions?
- MattPalmer1086 1y ago
  Mostly just integer factorisation of large numbers is hard. There are some other things you have to worry about practically, e.g Coppersmith's attack, and padding schemes (although that wasn't part of the proof I was given)

WhitneyLand 1y ago

Your comment about memorizing as part of understanding makes a lot of sense to me, especially as one possible technique to get get unstuck in grasping a concept. If it doesn’t work for you on l33t code problems, what techniques are you finding more effective in that case?

jdmichal 1y ago
I was part of an ACM programming team in college. We would review classes of problems based on the type of solution necessary, and learn those techniques for solving them. We were permitted a notebook, and ours was full of the general outline of each of these classes and techniques. Along with specific examples of the more common algorithms we might encounter. As a concrete example, there is a class of problems that are well served by dynamic programming. So we would review specific examples like Dijkstra's algorithm for shortest path. Or Wagner–Fischer algorithm for Levenshtein-style string editing. But we would also learn, often via these concrete examples, of how to classify and structure a problem into a dynamic programming solution. I have no idea if this is what is meant by "l33t code solutions", but I thought it would be a helpful response anyway. But the bottom line is that these are not common in industry, because hard computer science is not necessary for typical business problems. The same way you don't require material sciences advancements to build a typical house. Instead it flows the other way, where advancements in materials sciences will trickle down to changing what the typical house build looks like.
marcosdumay 1y ago
What I understand from the GP is that memorizing l33t code won't help you learn anything useful. Not that understanding the solutions won't help you memorize them.
simpaticoder 1y ago
>If it doesn’t work for you on l33t code problems, what techniques are you finding more effective in that case? Memorization of l33t code DOES work well as prep for l33t code tests. I just don't think l33t code has much to do with application programming. I've long felt that "computer science" is physics for computers, low on the abstraction ladder, and there are missing labels for the higher complexity subjects built on it. Imagine if all physical sciences were called "physics" and so in order to get a job as a biologist you should expect to be asked questions about the Schroedinger equation and the standard model. We desperately need "application engineering" to be a distinct subject taught at the university level.

bitexploder 1y ago

Memorizing is a super power / skill. I work in a ridiculously complex environment and have to learn and know so much. Memorizing and spaced repetition are like little islands my brain can start building bridges between. I used to think memorizing was anti-first principles, but it is just good. Our brains can memorize so much if we make them. And then we can connect and pattern matching using higher order thinking.

lvspiff 1y ago
Recognizing the patterns and applying patterned solutions is where I see success in my niche of healthcare interoperability. So much of my time is spent watching people do things,process and how they use data. It's amazing how much people remember to do their job, but me coming in and be able to bridge the doctor and the lab to share data easier is like Im an alchemist. It's really not a problem I've been able to see ai solve without suggesting solutions that are too simple or too costly and in that goldilocks zone everyone will be happy with
simpaticoder 1y ago
What's even better about memorization is that you have an objective method to test your own understanding. It is so easy to believe you understand something when you don't! But, at least with math, I think if you can reproduce the proof from memory you can be very confident that you aren't deluding yourself.
supriyo-biswas 1y ago
There's this [1] which is one of my favorite articles on that topic; definitely worth a read. [1] https://www.pearlleff.com/in-praise-of-memorization https://www.pearlleff.com/in-praise-of-memorization
hosh 1y ago
In education, I have heard it called “fluency”.

trod1234 1y ago

Mathematics pedagogy today is in a pretty sorrowful state due to bad actors and willful blindness at all levels that require public trust. A dominant majority in public schools starting late 1970s seems to follow the "Lying to Children" approach which is often mistakenly recognized as by-rote teaching but are based in Paulo Freire's works that are in turn based on Mao's torture discoveries from the 1950s. This approach contrary to classical approaches leverages torturous process which seems to be purposefully built to fracture and weed out the intelligent individual from useful fields, imposing sufficient thresholds of stress to impose PTSD or psychosis, selecting for and filtering in favor of those who can flexibly/willfully blind/corrupt themselves. Such sequences include Algebra->Geometry->Trigonometry where gimmicks in undisclosed changes to grading cause circular trauma loops with the abandonment of Math-dependent careers thereafter, similar structures are also found in Uni, for Economics, Business, and Physics which utilize similar fail-scenarios burning bridges where you can't go back when the failure lagged from the first sequence, and you passed the second unrelated sequence. No help occurs, inducing confusion and frustration to PTSD levels, before the teacher offers the Alice in Wonderland Technique, "If you aren't able to do these things, perhaps you shouldn't go into a field that uses it". (ref Kubark Report, Declassified CIA Manual) Have you been able to discern whether these "patterns" as you've called them aren't just the practical reversion to the classical approach (Trivium/Quadrivium)? Also known as the first-principles approach after all the filtering has been done. To compare: Classical approaches start with nothing but a useful real system and observations which don't entrench false assumptions as truth, which are then reduced to components and relationships to form a model. The model is then checked for accuracy against current data to separate truth from false in those relationships/assertions in an iterative process with the end goal being to predict future events in similar systems accurately. The approach uses both a priori and a posteriori components to reasoning. Lying to Children reverses and bastardizes this process. It starts with a single useless system which contains equal parts true and false principles (as misleading assumptions) which are tested and must be learned to competency (growing those neurons close together). Upon the next iteration one must unlearn the false parts while relearning the true parts (but we can't really unlearn, we can only strengthen or weaken) which in turn creates inconsistent mental states imposing stress (torture). This is repeated in an ongoing basis often circular in nature (structuring), and leveraging psychological blindspots (clustering), with several purposefully structured failings (elements) to gatekeep math through torturous process which is the basis for science and other risky subject matter. As the student progresses towards mastery (gnosis), the systems become increasingly more useful. One must repeatedly struggle in their sessions to learn, with the basis being if you aren't struggling you aren't learning. This mostly uses a faux a priori reasoning without properties of metaphysical objectivity (tied to objective measure, at least not until the very end). If you don't recognize this, an example would be the electrical water pipe pressure analogy. Diffusion of charge in-like materials, with Intensity (Current) towards the outermost layer was the first-principled approach pre-1978 (I=V/R). The Water Analogy fails when the naive student tries to relate the behavior to pressure equations that ends up being contradictory at points in the system in a number of places introducing stumbling blocks that must be unlearned. Torture being the purposefully directed imposition of psychological stress beyond a individuals capacity to cope towards physiological stages of heightened suggestability and mental breakdown (where rational thought is reduced or non-existent in the intelligent). It is often recognized by its characteristic subgroups of Elements (cognitive dissonance, a lack of agency to remove oneself and coercion/compulsion with real or perceived loss or the threat thereof), Structuring (circular patterns of strictness followed by leniency in a loop, fractionation), and Clustering (psychological blindspots).

raincole 1y ago

During my elementary school years, there was a teacher who told me that I didn't need to memorize it as long as I understand them. I taught he was the coolest guy ever. Only when I got late twenties, I realized how wrong he was. Memorization and understanding go hand in hand, but if one of them has to come first than it's memorization. He probably said that because that was what kids (who were forced to do rote memorization) wanted to hear.

HankStallone 1y ago
Probably. I hated memorization when I was a student too, because it was boring. But as soon as I did some teaching, my attitude changed to, "Just memorize it, it'll make your life so much easier." It's rough watching kids try to multiply when they don't have their times tables memorized, or translate a language when they haven't memorized the vocabulary words in the lesson so they have to look up each one.
- skydhash 1y ago
  There's things that you need to know (2*2 = 4) and there are things that you need to understand (multiplication rules). Both can happen with practice, but they're not that related. Memorization is more like a shortcut. You don't need to go through the problem solving process to know the result. But with understanding, you master the heuristic factors needed to know when to take the shortcut and when to go through the problem solving route. The Dreyfus Skill Model [0] is a good explanation. Novice typically have to memorize, then as they master the subject, their decision making becomes more heuristic based. LLMs don't do well with heuristics, and by the times you've nailed down all the problems data, you could have been done. What they excels at is memorization, but all the formulaic stuff have been extracted into frameworks and libraries for the most popular languages. [0]: https://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisition#The_Dreyfus_Skill_Model https://en.wikipedia.org/wiki/Dreyfus_model_of_skill_acquisi...
  - mason55 1y ago
    I think the problem is that in spots where the concepts build on one another, you need to memorize the lower level concepts or else it'll be too hard to make progress on the higher level concepts. If you're trying to expand polynomials and you constantly have to re-derive multiplication from first principles, you're never going to make any progress on expanding polynomials.

stonemetal12 1y ago

>Memorization is not a panacea. It is What you memorize that is important, you can't have a good discussion about a topic if you don't have the facts and logic of the topic in memory. On the other hand using memory to paper over bad design instead of simplifying or properly modularizing it, leads to that 'the worst code I have seen is code I wrote six months ago' feeling.

tshaddox 1y ago

> But an amazing thing happened - I realized that it was impossible to memorize a proof without understanding it! This may be true of mathematical proofs, but it surely must not be true in general. Memorizing long strings of digits of pi probably isn’t much easier if you understand geometry. Memorizing famous speeches probably isn’t much easier if you understand the historical context.

gundmc 1y ago
> Memorizing famous speeches probably isn’t much easier if you understand the historical context. Not commenting on the merits of critical thinking vs memorization either way, but I think it would be meaningfully easier to memorize famous speeches if you understand the historical context.
- codr7 1y ago
  Right, isn't building up a (imaginary) context how people memorize pi?
- tshaddox 1y ago
  For memorizing a speech word-for-word, I don't think so. Knowing the years of the signing of the Declaration of Independence and the Gettysburg Address aren't gonna help you nail the exact wording of the first sentence.

taeric 1y ago

I would wager some folks can memorize without understanding? I do think memorization is underrated, though. There is also something to the practice of reproducing something. I always took this as a form of "machine learning" for us. Just as you get better at juggling by actually juggling, you get better at thinking about math by thinking about math.

lazide 1y ago
Rote memorization is essentially that, yes.

Aurornis 1y ago

My controversial education hot take: Pointless rote memorization is bad and frustrating, but early education could use more directed memorization. As you discovered: A properly structured memorization of carefully selected real world material forces you to come up with tricks and techniques to remember things. With structured information (proofs in your case) you start learning that the most efficient way to memorize is to understand, which then reduces the memorization problem into one of categorizing the proof and understanding the logical steps to get from one step to another. In doing so, you are forced to learn and understand the material. Another controversial take (for HN, anyway) is that this is what happens when programmers study LeetCode. There’s a meme that the way to interview prep is to “memorize LeetCode”. You can tell who hasn’t done much LeetCode interviewing if they think memorizing a lot of problems is a viable way to pass interviews. People who attempt this discover that there are far too many questions to memorize and the best jobs have already written their own questions that aren’t out of LeetCode. Even if you do get a direct LeetCode problem in an interview, a good interview will expect you to explain your logic, describe how you arrived at the solution, and might introduce a change if they suspect you’re regurgitating memorized answers. Instead, the strategy that actually works is to learn the categories of LeetCode style questions, understand the much smaller number of algorithms, and learn how to apply them to new problems. It’s far easier to memorize the dozen or so patterns used in LeetCode problems (binary search, two pointers, greedy, backtracking, and so on) and then learn how to apply those. By practicing you’re not memorizing the specific problems, you’re teaching yourself how to apply algorithms. Side note: I’m not advocating for or against LeetCode, I’m trying to explain a viable strategy for today’s interview format.

bee_rider 1y ago
Memorization of, like, multiplication tables gives us a poor view of the more interesting type of memorization. Remembering types of problems we’ve seen. Remembering landmarks and paths, vs just remembering what’s in every cell of a big grid. I still don’t like leetcode, though.
- Aurornis 1y ago
  > Memorization of, like, multiplication tables gives us a poor view of the more interesting type of memorization. Memorizing multiplication tables is the first place many children encounter this strategy: The teacher shows you that you could try to memorize all of the combinations, or you could start learning some of the patterns and techniques. When multiplying by 5 the answer will end in 0 or 5. When multiplying by 2 the answer will be an even number, and so on.

dunham 1y ago

Fortunately that was not my experience in abstract algebra. The tests and homework were novel proofs that we hadn't seen in class. It was one of my favorite classes / subjects. Someone did tell me in college that they did the memorization thing in German Universities. Code-wise, I spent a lot of time in college reading other people's code. But no memorization. I remember David Betz advsys, Tim Budd's "Little Smalltalk", and Matt Dillon's "DME Editor" and C compiler.

jdougan 1y ago
Another advsys enjoyer! Did you ever write a game with it?

glitchc 1y ago

Indeed, not just math. Biology requires immense amounts of memorization. Nature is littered with exceptions.

johntarter 1y ago

Interesting I had the same problem and suffered in grades back in school simply because I couldn't memorize much without understanding. However, I seemed to be the only one because every single other student, including those with top grades, were happy to memorize and regurgitate. I wonder how they're doing now.

ghelmer 1y ago

I find it's helpful to have context to frame what I'm memorizing to help me understand the value.

eithed 1y ago

Depends on the subject - I can remember multiple subjects where the teacher would give you a formula to memorise without explaining why or where it came from. You had to take it as an axiom. The teachers also didn't say - hey, if you want to know why did we arrive to this, have a read here, no, it was just given. Ofc you could also say that's for the student to find out, but I've had other things on my mind

ian-g 1y ago

It's funny, because I had the exact opposite experience with abstract algebra. The professor explained things, we did proofs in class, we had problem sets, and then he gave us open-book semi-open-professor take-home exams that took us most of a week to do. Proof classes were mostly fine. Boring, sometimes ridiculously shit[0], but mostly fine. Being told we have a week for this exam that will kick our ass was significantly better for synthesizing things we'd learned. I used the proofs we had. I used sections of the textbook we hadn't covered. I traded some points on the exam for hints. And it was significantly more engaging than any other class' exams. [0] Coming up with novel things to prove that don't require some unrelated leap of intuition that only one student gets is really hard to do. Damn you Dr. B, needing to figure out that you have to define a third equation h(x) as (f(x) - g(x))/(f(x) + g(x)) as the first step of a proof isn't reasonable in a 60 minute exam.

__alexs 1y ago

Is it the memorisation that had the desired effect or the having to come up with the novel proofs? Many schools seem to do the memorising part, but not the creating part.

bwfan123 1y ago

My abstract algebra class had it exactly backwards. It started with a lot of needless formalism culminating in galois theory. This was boring to most students as they had no clue why the formalism was invented in the first place. Instead, I wished it showed how the sausage was actually made in the original writings of galois [1]. This would have been far more interesting to students, as it showed the struggles that went into making the product - not to mention the colorful personality of the founder. The history of how concepts were invented for the problems faced is far more motivating to students to build a mental model than canned capsules of knowledge. [1] https://www.ams.org/notices/201207/rtx120700912p.pdf https://www.ams.org/notices/201207/rtx120700912p.pdf

inetknght 1y ago
> This was boring to most students as they had no clue why the formalism was invented in the first place. > The history of how concepts were invented for the problems faced is far more motivating to students to build a mental model than canned capsules of knowledge. That's something I really like about 3blue1brown, and he says it straight up [0]: > My goal is for you to come away feeling like you could have invented calculus yourself. That is, cover all those core ideas, but in a way that makes clear where they actually come from, and what they really mean, using an all-around visual approach. [0]: https://www.youtube.com/watch?v=WUvTyaaNkzM https://www.youtube.com/watch?v=WUvTyaaNkzM

thoughtlede 1y ago

memorization + application = comprehension. Rinse and repeat. Whether leet code or anything else.