But I also abhor rote memorization. Not because I don't think it's valuable, I just hate extremely repetitive tasks. Close friends of mine with zero interest in the subject just took everything at face level and preferred to rote memorize everything and slap it down as needed. And they tested much better than I did. Neither I or they are more intelligent than one another and these days we all excel at our own areas we enjoy. But for that subject in particular, in a rushed classroom standardized testing situation, I think their approach was "better".
Once my engineering school load got too big for me to continue to use that approach in my math classes, I took an alternative strategy of just accepting theories without deeply understanding them and I had a much harder time applying those particular math concepts outside of those classes.
Knowing the range of application of a technique is important.
I further suspect that this is also the mechanism responsible for some people's apparent tendency to do all the talking in a conversation, rather than allow a balanced share of time between conversants. The quick decay of short-term information requires constant refreshing, and talking is one means of augmenting this.
I'm reminded of the classic Feynman story about discovering the different mental models that he/his coworkers used, and how those models affected thinking: https://youtu.be/Si6NbKqYEd8?t=105
The discussions on this post have been the first time I've really hard people describe this phenomenon in such clarity. It's really nice to know there are other people who are similar to me in this way, as I very very often feel like the "odd one out" in this regard...
I really agree: this discussion is great!
I elaborated as a reply to a sibling comment, but repeat after me: You need to memorize things in math. In my experience, there is no way around it, especially if you want full understanding. It is a tool.
I realized this when I saw one of the higher math geniuses in university, one who really understood most things better than most of us, learn equations and lemmas from flashcards one day.
That memorization may become superfluous after you worked with something for a while, but until then, it is not just tremendously useful, but downright necessary, to make an equation, or set of rules, or whatever, just "pop up" in your head while you're thinking something through.
Relatedly:
> I needed to deeply understand it at the most universally root level and create a painstaking mental model of it
That is my modus operandi in math. I want to fully understand it, down to every little detail. But in my experience, memorization is one step for that model to really build itself up, and stick.
Reading through the pages of a math book and going "oh, yeah, I totally understood that, neat" is useless if you later encounter a problem and go "huh, so, what was the exact equation of the Fourier transform again"?
And not just because you now have to look it up to apply it, but also because an equation for example is not just a jumble of symbols that you write down and fill in. It has structure, it has meaning. If you can recite it in your sleep, then you also immediately see properties of it when they are relevant.
As Andrew Wiles said: "Math is not a spectator sport."
It's hard to fully bring across what memorization does for math, but since I started just using a flashcard app (Anki) several years ago, I literally sometimes lie in bed at night, eyes closed and no notepad, and work through math in my head, trying to further understand some aspects. And because I can "look" at what I memorized in my head, it works really well.
There is no way to pass through these obstacles (without spending the multiple lifetimes it took to forge them from first principles) except to memorize them and gradually extend understanding backwards from that memorization into the broader context of dependencies that converge into its formalization.
But having this predicament explained to me up front, ideally somewhere around the age one learns about something as basic as fractions, would have been enormously helpful.
That is not the only reason, but also a big part of it, yes. Some of the things (but not all) that I memorize I admittedly don't actually understand. But usually, the sudden realization comes at a later point, when I understood more of something else.
Literally happened last week to me. I had been memorizing a "stupid" theorem for a while[1], not realizing why it's useful, until something I read was about discontinuities in the n-th derivation of a function, and what that means for the terms of the function's Taylor series, and it all lit up in me, tying it all together.
It's a good feeling!
[1] https://www.dsprelated.com/freebooks/sasp/Spectral_Roll_Off....
I am definitely a "hands on" math person. Almost every piece of math I learn has application ultimately, personally I would not really be motivated otherwise. I learn math because I want to use it. I then also find joy in the abstract beauty, but I would not uncover that beauty without something I could use it for in the first place in mind. For me specifically, a large application of the math I'm learning is signal processing (both digital and analog), and analog circuits (e.g. "EE stuff").
And in that, some of the learning of "meaningless abstractions" first, and later entangling it with actual application, definitely, positively helped in getting a good understanding of the underlying math, which I can in turn integrate with what I'm doing in the "real world".
Not-memorizing forces you to practice more and develop deeper intuition which helps you reconstruct info. Memorization is a side-effect or learning, not a cause.
Our brains are connection machines, not encyclopedias.
But I got there by, at first, "blindly" memorizing FS, FT, DFT, DTFT, and Laplace and z transforms. This allowed me to not only apply them to problems while, at the time, not being fully comfortable with them yet, but also, and this is really the biggest part of the point I'm trying to make, to "look at them" while I was either solving problems or just furthering my understanding of the field when, say, lying in bed before falling asleep.
To an extent that would not have been possible if I had to "refer back" to their definitions on paper.
It was a tremendous tool (that is what memorization is, a tool) to just be able to recite, say, the DFT, or the DTFT, even while my intuition was still gradually building up. And to internally look at the differences between DFT and DTFT. The same is true for the countless other little factoids in my arsenal of flashcards. I literally built a lot of math understanding, both abstract and of actual engineering problems I was working on, when sitting in my car, on my 45 minutes commute to work or back home.
> Our brains are connection machines, not encyclopedias.
And memorization is a tool for making new connections.
Memorization leads to a different kind of knowledge, I think. You don't know the thing, you know the steps to reproduce.
It's like my grandmother who knows several songs by heart, but if you ask about the second line in the third verse she'll sing it from the start until she reaches the line in question. For a song this is sufficient, because it is analogous to the primary use cases.
I really can't operate on things for which I don't have a deep intuition. If I really have to treat something as a blackbox, I need to explicitly spell out the assumptions and "compress them enough" (which is actually probably also just about building an intuition on those assumptions).
I envy people a lot that can just "copy & paste" some knowledge in their thinking.
I think I now got to the point where I accepted that to be a part of me and made it "my brand" and I see that I can add value with being different in this aspect.
I often wonder how it must feel like to be able to "copy paste" knowledge. If you painstakingly have to sweat through all kinds of things that others can just "copy&paste", it can feel like others are "cheating" - but in the end it's just me being jealous.
I dropped out of my masters in Statistics because after a point, I had questions and due to circumstances, couldn't find answers by myself and couldn't find help (this is before the days of youtube explosion).
Eventually through serendipitous circumstances, I enrolled in a computer course and realized that I no longer had this "why" problem anymore when it came to programming - I could get answers quickly AND it allowed me to express myself the way I wanted. Didn't turn back after that.
I wonder if intelligent people with poor short term memories are more impressed with LLMs than those with good short term memories because that was my immediate reaction and why it is basically unthinkable for me to unsubscribe from chatGPT4.
As a paired cybernetic system I gain enormously more from LLMs compared to myself with a good short term memory and paired.
I think the deep thinker/modeler's time has come. The non-deep thinker/modeler with great flash memory time has come and gone.
However I do believe long term memory is more robust so if you're in a mentally compromised state, such as having a headache, it will be less affected. So if you have to work through a hard problem on a deadline but have a headache, you can rely on heuristics stored in long term memory rather than deriving everything in short term memory.
Adding on, there's a definite weird trade off going on between memory and creativity that I think is very under-appreciated. I think memory acts as a bind on creativity, and it's really easy to make this intuitive with any number of examples whether it be youth or marijuana/psychedelics. When you reduce your ability to access memory you seem to gain an ability to explore new spaces. Likewise the more you learn and build structures in your brain around concepts, the harder it is to accept and process novelties. Feels like there's some similar trade offs going on as with this deep vs fast thinking.
At some point you must memorize in math.
By rote memorization, it is available in my head immediately, and I can use it to reason through things, without having to derive anything. It helps tremendously.
Then, at some point, it will become so familiar and understood through working with it, that the rote memorization becomes superfluous. For example, I've worked with the Laplace Transforms and the various Fourier Transforms (FS, FT, DFT, DTFT) so intensely by now, that I can absolutely just recite them by concept and understanding of why they are what they are.
But until then, rote memorization is basically a necessary tool to work with math.
The moment that made me realize that was when I saw a math genius at university, one who clearly understood the topics extremely well, going through his flashcards.
About three weeks into my first undergraduate class on abstract algebra, it dawned on me that the instructor wasn't giving me math tests. He was giving me vocabulary tests. In that class, most of the answers to questions flow straight from the definitions. Once I broke out the flashcards and started memorizing definitions, that class became almost trivial.
I used flashcards in all my classes after that to memorize terms, definitions, and concepts. Math and engineering are, for me anyway, like a foreign language. To converse in that language fluently, one must be very comfortable with the vocabulary. It just makes sense.
For me, memorizing math means just working on problems until the theorems and definitions are like muscle memory. I know other people get by on flash cards but I can’t stand them.
Heck, I just wrote a midterm in network flow theory today and they gave us a sheet with all the theorems and definitions in the course up to this point. Needless to say, memorizing that sheet wouldn’t help you at all on the proofs. You have to actually practice writing proofs to get good at it.
I'm not a mathematician, but I've seen a similar bifurcation in programming styles and preferences: something like "debuggers" vs "abstraction builders". Would you rather have less to learn up front at the expense of needing to track more details as you're working, or would you rather spend time learning or developing a conceptual foundation to reduce ongoing pressure on your working memory?
I figure this is why discussions about keeping programming "simple" are so unproductive: people end up talking past each other because one camp values reducing up-front complexity, the other reducing ongoing complexity, but everybody talks about it as if there's only one simple–complex dimension.
Those short memory people (who I think I may be among) that GP said about relate to which group of mathematicians and programmers?
I am one of those that prefer to reduce complexity and prefer lower level programming. I think that such a type of person likes C for example.
Systemic thinking - considering the system as a whole. Systematic thinking - considering the components of a system step-by-step.
Of course both are important at a global scale. But one person can focus on one side and make an contribution.
Ben Stiller do it meme
I need to derive things to understand them, ie music theory. I'm jealous of people who can memorize and take at face value but if I'm looking at a chord, I need to know the components that create that chord, which gets frustrating because music has a lot of rules that seem to be based on vibe and closeness. Two things can be identical but distinct based on their context.
It used to take me 2-3x as long to do homeworks or labs compared to other classmates. Same with work assignments. It often triggers an imposter syndrome type feeling.
Yet I have proof that I'm capable of solving complex problems, I understand certain things almost immediately compared to others, other things I need to study for a long time.
I tend to rarely know an answer on the spot, but I know how to determine many things, by knowing how to find the information needed.
I don't pretend to be a genius, but I have proof via a degree, others opinions of me, and material results that basically say I'm intelligent to a point.
Once I get into a flow I can retain a fairly complex system in my head but before or after that state, it's a terrible blur where I can barely focus my eyes.
You are far smarter than you think.
Learn to get into flow, and also figure out how to get your state back faster... Once you do, it is a super power.
Simple things like leaving your editor/IDE open, create the same situation you left. I personally use music + IEMs to help me block sound and focus. -30-35 DB off the IEMs, and +70DB off my music... yeah, I don't hear anything.
I learned the editor trick recently, though I'd probably done it un-intentionally for years. I stopped, and I was wondering what the inertia I was facing was. It is only 2-3m to get setup... that's 2-3m to lose my thoughts, etc.
Good luck.
For me the best way to get into the flow is by directly getting back to where I left. If I take time to wake up, make breakfast or anything else I usually lost my ideas already and it takes me hours to get back to them.
Just writing it down, will help you remember. If not: You have the paper.
This is underrated. And, specifically...writing it down, with a pen or pencil. Typing it doesn't seem to have the same effect, even with some ritual of typing it into a fresh/separate document.
I used to use whatever cheap ballpoint was around, and I had terrible handwriting. But on a friend's recommendation I tried a fine point gel pen, the Uniball Signo DX 0.38mm. It's such a joy to write with and my handwriting is orders of magnitude more legible.
There's apparently a whole world of pens (and pen enthusiasts) out there: https://www.penaddict.com/top-5-pens
The only other software I usually have running on my work computer is a web browser.
People are always blown away by my ability to recall a story perfectly or create 2 page white boarded proofs, but inability to remember my SSN or why I entered a room.
A few years back I accepted a challenge to take a year off drinking from a family member.
I cannot tell you how much it improved my memory.
Before that I was always losing things, frustrated I couldn’t remember the names of things etc.
After about 3 months I felt like I had doubled my memory.
I wasn’t really a heavy drinker, but I would go for a few months having 1-2 beers a day.
Anyway just curious if you drink and have tried taking a break?
A related theory: being good at math, especially mental math, correlates with aphantasia (not being able to see pictures in your head). People who can see images in their head learn to do arithmetic early on with visual algorithms, which are fundamentally not good for understanding as well as rather error-prone (because the brain remembers gestalts, not finicky details like where a decimal is).
Aphantasiacs are forced to learn to do math differently, and use some different part of the brain as 'scratch space'. In my case it's the language brain: calculations which are set aside live in the same part of your brain that can repeat what was said a moment to you without understanding it. Turns out, though, that this verbal part is quite _accurate_ at remembering things, and this makes it easier to juggle multi-step calculations without paper.
This is the kind of condition I wonder if I have. Because on one hand, if I do a visualization exercise I would describe it as foggy at best. Then again, I struggle to understand how our brains could be wired that differently from person to person, that we could have or lack certain mental senses. I would suggest that everyone has some latent capability, whether they recognize it or not. After all the visual cortex is quite important.
In the context of your example, those good at mental math might see a chalkboard like image of the problem with all the detail. Those who struggle might be distracted visualizing the items that are being counted themselves, and all their detail.
For another riddle, consider the question of whether one recognizes their thoughts as an internal monologue or not, and how that relates to communication and action.
I agree, though, that visualization is probably latent even in people who can't do it. I notice it when half asleep for instance.
Experiments with chess players and memorization shed some light on this.
DeGroot’s research showed the average person is quite poor at recalling chess positions from real games, while chess masters have almost perfect recall even after seeing the board for only a few seconds.
However, when the chess masters were given completely random arrangements of pieces on the board (as opposed to positions from real chess games), they were no better than the average person.
This and other experiments suggest it’s not a visual snapshot, but functional knowledge and chunking, such as knowing pawns typically protect one another in diagonal chains, etc.
I recall from talking with high rated chess players, they rarely visualize the board in the mind’s eye, but they just know things without thinking, like, a bishop on d3 attacks the square h7 and not h6. Not from rote memorization, but from playing and seeing that sacrificing a bishop on h7 is a thematic attacking pattern.
as someone who used to do math competitions as a kid, everyone I knew who was "good" at mental math had a good number sense, a big bag of tricks for doing different things with numbers, and practiced with their bag of tricks.
A lot of this stuff seems highly related to ADHD. I exhibit them, and have done quite some amateur research myself on these behaviors and others seem to relate as well.
P.S: I am also an i3wm user btw ;)
I am working on a personal knowledge system to compensate. What are your ideas on this subject?
Popular PKS systems expect the user to adapt to "the system," rather than adapting to the user's particular set of needs. I consider each user to be a unique combination of common components.
I am more comfortable with deep thinking, or reflection, which allows me to come up with more complex analysis, while this process translates quite well when I have homework challenges.
I used to test some of my mental capacity with pseudo-IQ tests and realized that the factor that would penalize me the most was lack of time.
If you do a quick search for "IQ vs working memory", all the results show a positive correlation.
On the other hand, if my hypothesis was true, we would have discovered it already, so it's probably wrong.
It also may well be that the opposite is true — STM is prioritized by people who haven’t learned to properly recruit for profound structures. Speed may well have been a survival advantage a million years ago, but it’s not clear that this hasn’t changed since.
We also don’t know if the computing model of the brain is even appropriate at all. Humans have always assumed that the latest state of technology can represent the brain: clay and waterworks, mechanics, computers, etc.
At the end of the day, we don’t know how much we actually know about how the brain works because we don’t know what we don’t know…
In other words if you have a weak short-term or working memory you have to, by necessity, deeply think through everything - whether simple or complex. This may allow spotting the rare inconsistency or opportunity others may not, but as it comes at a performance and time penalty which, under circumstances commonly encountered in one's educational career (timed examinations) it more typically results in filtering such people out of fields where excursive, highly lateral modes of thinking would be beneficial.
To survive this filter one must either become (or be gifted with) the talent of exceptionally swift traversal of a large and heterogeneous general knowledge graph, as one cannot rely on a handful of tightly knit but relatively isolated silos of memorized specialization.