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I elaborated as a reply to a sibling comment, but repeat after me: You need to memorize things in math, especially if you want full understanding. It is a neces

by anyfoo 3y ago

I elaborated as a reply to a sibling comment, but repeat after me: You need to memorize things in math, especially if you want full understanding. It is a necessary 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 axioms, theorem, or whatever, just "pop up" in your head while you're thinking something through. You say: > 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 actually 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, and are able to make further connections. 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.

hgsgm 3y ago
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.
- anyfoo 3y ago
  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.
ChainOfFools 3y ago
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.
- hgsgm 3y ago
  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.
  - anyfoo 3y ago
    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".

by anyfoo 3y ago

hgsgm 3y ago

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.

anyfoo 3y ago
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.

ChainOfFools 3y ago

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.

hgsgm 3y ago
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.
- anyfoo 3y ago
  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".