I took your approach to math and it was vital to my scholastic success.

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.

i imagine this is probably the healthier/much better approach to learning anything. unfortunately society mostly prioritizes extreme speed and shallow understanding of anything due to the upside of moving quickly and making numbers go up at the end of a quarter.

I discovered the difference in these two types of students much later than I would have liked: last year of college. I like you, felt like I really needed to understand the material to do well, and I did, but there were so many students at discussion sections and office hours with much shallower understandings who would perform better on exams and assignments. These students often ask questions like, “is it always the case…” looking for a hammer they can apply to all similar problems rather than trying to understand nuances. Learning how to get an A in the course is very different than learning the course material.

Are you sure that the people who learned the material are dumber than you?

Knowing the range of application of a technique is important.

Just a hunch, but do you find that you perform substantially better if you are an environment where you can talk to yourself out loud as you work through a math or physics problem? If so, this is very much indicative of the phenomenon I'm describing, because what's happening is you're recruiting your audio memory as a compensatory short-term workspace, being able to loop some 2 or 3 additional elements of information in this extra area essentially acts as an extension of your working store. And guess what behavior students are typically prohibited or highly discouraged from doing during an exam?

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.

Actually for me I perform best when I can hear my inner monologue. If I'm in a social setting learning math it's not ideal at all, because I can't "listen" to myself think: I think through things entirely in my head when I can reason through and almost dictate to myself. But not verbally! It's almost like the moment I try to verbalize anything it gets muddied by whatever weird social/colloquial construct I've made to communicate with others.

That's really interesting! Do you know any other techniques to hijack our brains for performance boosts?

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

I write. It’s like having an internal monologue slowly repeating you whatever you’re thinking so it let’s the brain re-structure knowledge. It’s super helpful to force myself to pay attention. I rarely read what I wrote but because I did, there’s a higher chance I remembered.

Yeah, I basically have no choice but to deeply, truly understand something. If I don't "completely get it", I will struggle endlessly to employ the "knowledge" I have. It's been a lifelong challenge (indeed I barely finished high school because I was actually failing classes because it was impossible to keep up -- yet I am a senior software engineer today), while some people forge ahead seemingly effortlessly, I am still solidifying the fundamentals. To be fair though, my deep understanding of stuff results in pretty comprehensive/effective solutions that consider edge-cases/gotchas/etc. and I think that's pretty valuable. Once I have those "fundamentals", I have mastered that shizzle and it's completely solid in my repertoire basically forever. The decay is VERY slow once I learn something, and I can remember deep details about things for years.

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 pretty much have the same experience as you! (See my other comment in this thread)

I really agree: this discussion is great!

In my twenties I decided not to get a motorcycle for this very reason. I realized I would have disappeared for six months, learning every aspect of every part. And there were other things in my life that needed doing.

> 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.

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.

I agree with this, and the rationalization that I eventually worked out and learned to live with is that as you get into more advanced areas of math, much of what you are learning are building blocks for assembling more complex tools - but those are tools you have no use for yet, and therefore can't hook constituent elements into any existing framework of understanding.

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.

> except to memorize them and gradually extend understanding backwards

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....

No! Instead of rushing ahead into meaningless abstraction, you should spend more learning in the world of concrete examples and applications, to provide meat for those abstract skeletons. We don't need a mathematician is who is a poor imitation of a computer or a reference book.

In my experience, it helps to approach from both sides, and that's also part of the point I'm trying to make. If you come only from the "concrete examples and applications" side, you might not actually fully understand, despite thinking you do, and miss some finer subtleties. Those are often the "gotchas" that get pointed out in textbooks (but even then are easily forgotten).

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".

The OP article is in part about how people whom memorize things perform worse because their memory is incorrect, while people who understand things will spend more time making sense of a problem of solution instead of more quickly guessing part-blind.

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.

I think that I truly, positively understand both the Fourier transforms I deal with (FS, FT, DFT, DTFT, as well as their subtle but very important differences), and the Laplace transform and the z-transform. I can recite them just by "concept", that means by knowing what they do, how they work, and why they work, I can piece together the equations in no time. Memorization is not necessary anymore.

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.

> I also abhor rote memorization. Not because I don't think it's valuable, I just hate extremely repetitive tasks.

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.

Fully agree. I am annoyed by the educational pendulum swinging so far away from rote memorization that it eliminates it altogether.

I had a very similar experience studying physics and I still have some form of that in my current job as software engineer working on a quite large code base:

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.

This is like reading my life story.

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.

My short term memory sucks too. I need many exposures to retain the information for a standardized test.

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.